Method and apparatus to reduce variation of excess fiber length in buffer tubes of fiber optic cables

ABSTRACT

The present invention provides a method for reducing and/or controlling the variations of excess fiber length along the length of reeled fiber optic buffer tubes during the manufacture of the buffer tubes. The present invention varies any number, or combination, of parameters during the manufacture of buffer tubes to achieve a substantially uniform excess fiber length along a reeled buffer tube. One embodiment of the inventive method uses monotonically decaying draw or take-up tension of the buffer tubes during winding, combined with a stiffness-compliant pad placed on the reel core to aid in providing a substantially uniform excess fiber length in the tube, while another embodiment uses a monotonically increasing angular speed of the reel in combination with the stiffness-compliant pad on the reel core. In yet another embodiment a pad is placed either periodically or continuously in the windings of the buffer tube to provide an absorbing layer for the residual stresses existing in the buffer tube as it is reeled and after the reeling is complete, combined with re-reeling the buffer tube onto a second reel after the buffer tube has cooled (after manufacture), where the pad is removed during the re-reeling process. Additionally, the present invention can have the layers of buffer tube separated with rigid, cylindrical panels separating the layers. The present inventive method also combines any, or all, of the above steps to aid in achieving a substantially uniform excess fiber length along the length of the reeled buffer tube.

This application claims benefit of U.S. Provisional Application No.60/256,454 filed on Dec. 20, 2000, under the provisions of 35 USC119(e).

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention generally relates to the field of optical fibers,in particular to the manufacture of fiber optic buffer tubes having asubstantially constant excess fiber length (“EFL”) ratio throughout thelength of the buffer tube.

2. Discussion of Related Art

Optical fibers are very small diameter glass strands which are capableof transmitting an optical signal over great distances, at high speeds,and with relatively low signal loss as compared to standard wire orcable (including wire cable) networks. The use of optical fibers intoday's technology has developed into many widespread areas, such as:medicine, aviation, communications, etc. Most applications of opticalfibers require the individual fibers to be placed into groupings, suchas in fiber optic cables.

There are many ways to manufacture and configure fiber optic cables. Oneof the most common methods of manufacturing a fiber optic cable is byplacing a number of fiber optic buffer tubes in a single cable jacket,where each of the buffer tubes is a separate tube having a number ofindividual optical fibers or optical fiber ribbons. The buffer tubesthemselves are hollow tubes generally made from thermoplastic materials.

When a cable construction uses a number of buffer tubes, each containinga number of fibers, the quality of the finished cable greatly depends onthe quality of the components it uses, in particular, the buffer tubes.The quality of the individual buffer tubes can be affected by a largenumber of factors, and the manufacture of the buffer tubes, with thefibers, is one of the most critical of these factors. A common method ofmanufacturing the buffer tubes is to draw the tubes with the fibersplaced inside of the tubes. The buffer tube is then wrapped around aspool and left to cool at room temperature. During this process thetubes are reeled (in the drawing process) onto a hard or rigid spool(made of any sturdy material for example, wood or steel), and are drawnat a constant draw or take-up tension (on the tube itself) and with aconstant angular velocity of the spool.

However, when reeling of the buffer tubes occurs with constant tensiledraw on the tubes and constant angular speed of the spool the result isnon-uniform distribution of residual stresses along the length of thebuffer tube as it sits on the spool. In some cases, the non-uniformdistribution of EFL remains after taking the tube off the spool andsubsequently negatively affecting attenuation in the finished cable. Themain components, or origins, of the non-uniform, along the tube length,residual deformation stem from (1) stresses along the reeled buffer tubeaxis (i.e. circumferential stresses) which is a function of the distancefrom the reel surface, i.e. current radius and (2) in a transversedirection (i.e. radial stresses) typically varying from zero on the rollsurface to a maximum amount on the reel surface. It is noted that thisproblem not only exists in the fiber optic industry, but also in anyother industry where the extended rolling of a product is required. Forexample, the same problems exist in the manufacture of paper, electricalcable, aluminum sheet, etc. Resulting from these non-uniform stresses onthe spool is the permanent or residual deformation of the rolledmaterial (i.e. buffer tubes) and the creation of residual strains in thetubes and the fibers within the tubes. The creation of residual strainin the fibers is a very serious problem in the manufacture of fiberoptic cables and buffer tubes that causes variation of EFL along thelength of the tubes and subsequently attenuation problems.

Excess Fiber Length (“EFL”) is an important parameter affecting thequality and performance of a fiber optic cable. EFL is generally definedas a relative difference between the actual individual fiber length(defined as “L_(F)”) and the length of the buffer tube from which thefiber came (defined as “L_(B)”), where the %EFL=[(L_(F)−L_(B))/L_(B)]×100. EFL is important in the proper operationof a fiber optic cable. In general, it is desirable to have a smallpositive EFL. This means that the length of the fibers is larger thanthe length of the buffer tube in which the fibers are disposed. Thisadded length allows delayed stretching of the buffer tube under atensile load during installation or its use without adding any tensileloads on the fibers meaning that, to a certain level of tensile load,the load will be carried by the strength rods or tapes, and not involvethe fibers. However, it is important that the EFL should not be toolarge and have a relatively even distribution throughout the length ofthe buffer tube. When the EFL distribution throughout the length of atube is non-uniform it can adversely affect the operation and efficiencyof the cable as a whole.

Through testing, it has been discovered that measurements of EFL inbuffer tubes wrapped according to prior art methods results in buffertubes which show an EFL distribution having a skewed parabolic shape.This is depicted in FIG. 1, which shows a graphical representation of atypical EFL distribution in a buffer tube wound under the prior artmethodology. As shown, the EFL curve 1 is of a typical buffer tubelength after manufacture. The left side of the graph indicates the EFLat the early stages of the tube manufacture (i.e. the beginning portionof the tube on the reel). The rapid or steep change in the left part ofthe EFL curve occurs in the tube length during the initial wraps of thetube on the reel. The remaining portion of the curve shows that the EFLpeaks at some point near the center of the tube length and then tapersoff near the end of the tube winding.

This is a significant problem with long buffer tubes (approximately 10km in length) and having a relatively small core radius for the take-upreel (around 100 mm). When the parabolic variation becomes too large thefiber attenuation near the middle of the length of the cable can besignificant, thus making the cable useless.

An additional problem of the prior art methods of manufacturing buffertubes is the limiting effect on the line speeds of the manufacturingprocess due to the uneven EFL distributions. As line speeds increase theEFL distribution problems become more significant. Therefore, to avoidthese problems, manufacturing speeds are limited so as to preventsignificant EFL problems.

SUMMARY OF THE INVENTION

The present invention is directed to eliminating or greatly reducing theimpact of the above problems by the use of prior art methods ofmanufacturing buffer tubes with optical fibers.

The present invention uses the variation of a number of differentparameters or physical characteristics of the manufacturing process orequipment, either individually or in combination, to provide asubstantially uniform EFL distribution along the entire length of amanufactured buffer tube. In a first embodiment of the presentinvention, a pad having a compliant stiffness is placed on the core ofthe take-up reel prior to the winding of the manufactured buffer tube,and the take-up tension of the tube as it is being drawn ismonotonically decayed according to a set function so as to ensure aneven EFL distribution along the entire length of the buffer tube.Although it is contemplated that the present invention can provide aneven EFL distribution without the use of a stiffness-compliant pad, itis to be used in the preferred embodiment to provide stress relief inthe initial layers of the buffer tube, located closest to the core.

In a second embodiment of the present invention, a substantially evenEFL distribution is accomplished by using a combination of the stiffnesscompliant pad on the core of the reel with varying the angular speed ofthe take-up spool during the spooling of the buffer tube. Similar to thefirst embodiment, the variation in the speed of the take-up spoolcombined with the stiffness-compliant pad is used to provide asubstantially even stress and strain distribution throughout the lengthof the tube, thus resulting in a substantially even EFL distribution. Aswith the first embodiment, it is contemplated that only the variation inthe angular speed of the take-up spool can be used to provide an evenEFL distribution, but in the preferred embodiment the combination is tobe used.

In a third embodiment of the present invention, stiffness-compliant padsare placed between layers of tube windings at intervals throughout thereeling of the buffer tube, as well as on the spool core. The use ofthese pads at intervals allow the excess stress and strain in the tubesand fibers to be absorbed in the pads. It is preferred, in thisembodiment, that the use of the stiffness-compliant pads be combinedwith either varying the take-up tension or the angular speed of thespool, as discussed in the previous two embodiments. Further, in thisembodiment, pads can be placed between each winding or at regularintervals in the tube winding. Additionally, this embodiment can be usedwith a stiffness-compliant pad on the reel core, as described above. Itis preferred in this embodiment, that the tubes be re-reeled after theinitial reeling step and the tube is allowed to cool to room temperatureto aid in achieving a more uniform EFL distribution. The re-reeling stepcan be used with any of the above embodiments.

In the fourth embodiment, the layers are separated with rigid,preferably metal or composite cylindrical panels separating the layersand thus “breaking” up the stress compounding from upper layers. Thepanels can have slots to allow the tube to continue onto the next level.

It is to be noted that it is further contemplated that although theabove embodiments can be used individually to obtain a substantiallyeven EFL distribution, it is contemplated that any combination of theembodiments, or components thereof, can be used without altering thescope or spirit of the present invention. For instance, monotonicallydecaying the take-up tension may be combined with the varying of theangular spool speed and the use of stiffness-compliant pads or stiffcylindrical separators in the windings of the spool to achieve an evenEFL distribution.

BRIEF DESCRIPTION OF THE DRAWINGS

The advantages, nature and various additional features of the inventionwill appear more fully upon consideration of the illustrativeembodiments of the invention which are schematically set forth in thedrawings, in which:

FIG. 1 is a graphical representation of a typical parabolic residualdistribution of EFL along the length of a buffer tube;

FIG. 2 is a graphical representation of the residual distribution of EFLalong the length of a prior art buffer tube, along with a distributionof a buffer tube made in accordance with the present invention;

FIG. 3 is a diagrammatical representation of a buffer tube manufacturingapparatus;

FIG. 4A is a diagrammatical representation of a thick walled cylinderused in an analytical model during the development of the presentinvention;

FIG. 4B is a diagrammatical representation of a single model element ofthe thick walled cylinder shown in FIG. 4A;

FIG. 4C is a diagrammatical representation of boundary conditions of theelement shown in FIG. 4B;

FIG. 5A-1 is a graphical representation of the distribution ofcircumferential stress for 10 and 50 buffer wrappings of a thick walledcylinder as shown in FIG. 4A, along the roll radius;

FIG. 5A-2 is a graphical representation of the distribution of radialstress for 10 and 50 buffer wrappings of a thick walled cylinder asshown in FIG. 4A, along the roll radius;

FIG. 5A-3 is a graphical representation of the distribution ofcircumferential and radial stress for 10 and 50 buffer wrappings of athick walled cylinder as shown in FIG. 4A, along the roll radius;

FIG. 5B-1 is a graphical representation of the influence of maximum rollradius on the distribution of compressive stresses for a roll with 460wraps; 2500 wraps and 5000 wraps;

FIG. 5B-2 is a graphical representation of the influence of maximum rollradius on the distribution of compressive stresses for a roll with 50000wraps;

FIG. 6-1 is a graphical representation of the distribution ofcircumferential and radial strains for 10 wraps along the roll radius ofa wrapped thick walled cylinder with a constant reeling take-up stress;

FIG. 6-2 is a graphical representation of the distribution ofcircumferential and radial strains for 50 wraps along the roll radius ofa wrapped thick walled cylinder with a constant reeling take-up stress;

FIG. 6-3 is a graphical representation of the distribution ofcircumferential and radial strains for 460 wraps along the roll radiusof a wrapped thick walled cylinder with a constant reeling take-upstress;

FIG. 7-1 is a graphical representation of EFL along the roll radius of awrapped buffer tube for 10 wraps, and 460 wraps with a constant reelingtake-up stress;

FIG. 7-2 is a graphical representation of EFL along the roll radius of awrapped buffer tube for 2500 wraps with a constant reeling take-upstress;

FIG. 8 is a graphical representation of a parabolic decay in tensiletake-up stress at a constant tensile load, a slowly decaying tensileload and a rapidly decaying tensile load;

FIG. 9A-1 is a graphical representation of circumferential stressdistribution for 10 wraps along the roll radius using an analyticalmodel;

FIG. 9A-2 is a graphical representation of radial stress distributionfor 10 wraps along the roll radius using an analytical model;

FIG. 9A-3 is a graphical representation of the circumferential andradial stress distributions for 10 wraps along the roll radius using ananalytical model;

FIG. 9B-1 is a graphical representation of the circumferential andradial stress distributions for 10 wraps along the roll radius using twodifferent analytical models;

FIG. 9B-2 is a graphical representation of the circumferential andradial stress distributions for 2460 wraps along the roll radius usingtwo different analytical models;

FIG. 9C-1 is a graphical representation of the circumferential andradial strain distributions for 10 wraps along the roll radius using twodifferent analytical models;

FIG. 9C-2 is a graphical representation of the circumferential andradial strain distributions for 2460 wraps along the roll radius usingtwo different analytical models;

FIG. 10 is a graphical representation of proposed variable angularvelocities to control the EFL distribution in rolls of buffer tubes onsoft and rigid spool cores;

FIG. 11 is a diagrammatical representation of the relationship betweenTensile Stress, Shrinkage, Time on the Reel and Creep of a buffer tube;

FIG. 12 is a graphical representation of shapes of circumferential andradial stresses as compared to roll radius with varying parameters;

FIG. 13 is a tabular representation of shapes of circumferential stresscurves as a function of the parameters α (Decay in the Take-up Stress)and β (Core or Pad-on-the-core Stiffness);

FIG. 14A is a diagrammatical representation of a finite element meshmodel for an initially coiled model used in an analysis of the presentinvention;

FIG. 14B is a diagrammatical representation of the finite element meshmodel shown in FIG. 14A after deformation under a load;

FIG. 14C is a graphical representation of circumferential and radialstress distributions of the model in FIG. 14A under a load;

FIG. 15A is a diagrammatical representation of a finite element modeland mesh used for a dynamic winding simulation in the development of thepresent invention;

FIG. 15B is a graphical representation of curves representing theapplication of the tension and angular velocity for the dynamic windingmodel shown in FIG. 15A;

FIG. 15C is a graphical representation of circumferential and radialstress distributions of the model in FIG. 15A under a load;

FIG. 15D is a graphical representation of axial strain over the lengthof the model in FIG. 15A under a load;

FIG. 16 is a graphical representation of circumferential and radialstresses in the dynamically wound sheet shown in FIG. 15A, for a slowerloading rate case;

FIG. 17 is a graphical representation of circumferential strain over thelength of the model in FIG. 15A for fast and slow loading rate dynamicwinding cases;

FIG. 18 is a graphical representation of a comparison of circumferentialand radial stresses obtained using the a finite element model and ananalytical model, for 10 wraps at different loading rates;

FIG. 19A is a diagrammatical representation of a finite element mesh fora concentric layer model of a wound buffer tube;

FIG. 19B is a graphical representation of radial and circumferentialstress distributions for the concentric layer model in FIG. 19A;

FIG. 19C is a graphical representation of radial and circumferentialstress and strain distributions through the radius of the concentriclayer model shown in FIG. 19A;

FIG. 19D is a graphical representation of circumferential strain and EFLdistribution for the model shown in FIG. 19A under constant tensionduring manufacture;

FIG. 20 is a graphical representation of radial and circumferentialstress and strain distributions for different values of elastic modulusfor the model shown in FIG. 19A;

FIG. 21 is a graphical representation of radial and circumferentialstress and strain distributions for different core diameters of themodel shown in FIG. 19A;

FIG. 22 is a graphical representation of radial and circumferentialstress and strain distributions for different values of constant tensionapplied to the model shown in FIG. 19A;

FIG. 23 is a graphical representation of radial and circumferentialstress and strain distributions for different levels of linearlydecaying tension applied to the model showing in FIG. 19A;

FIG. 24 is a graphical representation of radial and circumferentialstress and strain distributions for different types of compliant layerson the core surface in the model shown in FIG. 19A;

FIG. 25A is a graphical representation of radial and circumferentialstress and strain distributions for different cases of distributedcompliant layers in the model shown in FIG. 19A;

FIG. 25B is a graphical representation of radial and circumferentialstress and strain distributions for different cases of distributed stifflayers in the model shown in FIG. 19A;

FIG. 25C is a graphical representation of radial and circumferentialstress and strain distributions for different cases of internal pressurein the model shown in FIG. 19A;

FIG. 26A is a graphical representation of radial and circumferentialstress and strain distributions for a compliant layer combined withlinearly decaying tension in the model shown in FIG. 19A;

FIG. 26B is a graphical representation of radial and circumferentialstress and strain distributions for a compliant layer combined withlinearly decaying tension with varying values for Young's Modulus of themodel shown in FIG. 19A;

FIG. 27A is a graphical representation of tension curves for differentvalues of the coefficient ax in the model shown in FIG. 19A;

FIG. 27B is a graphical representation of radial and circumferentialstress and strain distributions for a compliant layer combined withlinearly and non-linearly decaying tensions in the model shown in FIG.19A;

FIG. 28 is a graphical representation of radial and circumferentialstresses obtained from a finite element layer model and an analyticalmodel;

FIG. 29 is a diagrammatical representation of buffer tube packing on areel;

FIG. 30 is a graphical representation of EFL distributions obtained froma finite element analysis and through experimentation for a constanttension;

FIG. 31 is a graphical representation of EFL distributions obtained froma finite element analysis and through experimentation for a lowerconstant tension than that applied in FIG. 30;

FIG. 32A is a graphical representation of EFL distributions obtainedfrom a finite element analysis and through experimentation for twodifferent cases of constant tension;

FIG. 32B is a graphical representation of EFL distribution obtained froma finite element analysis and through experimentation for the case of acompliant layer on the reel surface and variable tension applied to abuffer tube;

FIG. 33 is a graphical representation of the distribution of EFL inthree different buffer tubes;

FIG. 34 is a graphical representation of variation of EFL in two buffertubes, wherein one tube is reeled without a pad on the reel, and atconstant take-up tension, and one tube is reeled with a monotonicallyreduced take-up tension on a reel with a thick soft pad;

FIG. 35 is a graphical representation of variation of EFL in two buffertubes, where on tube has a rigid reel core and constant take-up tension,while the other has a thin foam layer on the core and decaying take-uptension;

FIG. 36 is a graphical representation of linear speed of a tube as afunction of time;

FIG. 37 is a graphical representation of three different EFLdistributions as a function of length of the buffer tube and variationsin angular speeds;

FIG. 38 is a diagrammatical representation of a buffer tubemanufacturing system according to one embodiment of the presentinvention;

FIG. 39 is a graphical representation of EFL in two buffer tube samplesafter loading with a parabolically decaying take-up load;

FIG. 40 is a diagrammatical representation of a pad or stiffness memberbeing inserted into a buffer tube winding according to one embodiment ofthe present invention; and

FIG. 41 is a diagrammatical representation of an apparatus to be used toperform the present invention.

DETAILED DESCRIPTION OF THE INVENTION

The present invention will be explained in further detail by makingreference to the accompanying drawings, which do not limit the scope ofthe invention in any way.

In the development of the present invention, a significant amount oftesting and experimentation was conducted. The goal of this research wasto better understand the cause of the EFL parabola, and to find a meansto control and flatten the EFL parabola through a “smart process”. As afinal goal, the EFL distribution along the length of buffer tube shouldbe a small constant, preferably from approximately 0.05% to 0.15%.

In order to achieve the main research goal, it was necessary to modelthe winding process for buffer tubes (see FIG. 3 showing a buffer tube30 under tensile force “T” reeled on a spool “S” that rotates at angularvelocity “ω”) and to analyze the stress distribution as a function ofgeometrical, process and material parameters. This was done using twomain analytical tools, the first being analytical closed form solutionsand the second being finite elements analysis, the results of which werethen compared to experimental results. The goal of the analyticalclosed-form solutions was to obtain equations relating the variation inthe EFL, strain and stress to the core radius, maximum radius of theroll, material properties and take-up tension. The goal of the finiteelement modeling was to better understand the influence of materialparameters and the transient dynamic process (continual adding ofstressed material to an already stressed deformed body), and to considerinstability of motion and sliding of layers with friction.

The desired outcome was to find a method and apparatus which allowed themanufacture of buffer tubes with relatively small EFL variationthroughout the length. Such a distribution is shown in FIG. 2, where thedistribution 2 is a substantially even EFL distribution, throughout thelength, achieved by the present invention.

Based on preliminary analysis, it was suggested that the parabolicdistribution of EFL was a result of the parabolic distribution incircumferential stress. Consequently, the solution to the problem ofstraightening the EFL parabola was determined to include straighteningthe circumferential stress curve.

In order to verify these assumptions, the research program includedsimplified analytical solutions, refined finite element models, andexperiments to analyze the process of winding of buffer tubes. Thecontribution of major factors to the distribution of stresses in thewound material was analyzed. The first analytical model considered was athick-walled cylinder under tensile stress. The second analytical modelwas based on the model of a shrunk ring; this model additionally tookinto consideration the difference between the stiffness of the wrappedmaterial and that of the reel core. Also, the second analytical modelallowed for the consideration of a variable winding tension, whichaccording to the present invention, is one method used to create asubstantially uniform EFL distribution.

The buffer tube roll geometry was characterized by initial radius, i.e.radius of the reel core, and the maximum radius of the roll. The maximumradius was expressed in terms of the number of wraps and thickness ofthe material (such as thickness of a tape or outer diameter of a buffertube). The take-up tension was represented by a wrapping stress, whichmay or may not vary during the winding process. Considering theequations of equilibrium in terms of stresses provided a short andsimple way to compute the stress distribution in the wound roll.

In order to compute the distribution of strains in the roll, additionalequations relating stresses and strains through material properties wereconsidered. The stresses and strains were related to each other throughthe equations of plane stress or plane strain.

In the computation of EFL, it was assumed that the fabrication processresulted in a close to constant value of EFL_(o) (which is the EFL ofthe tube after the tube is manufactured but prior to the tube beingreeled). This value, however, undergoes changes because of a non-uniformstress field in the structure on the reel. The optimization goal is tominimize the range of variation of EFL in the roll and to adjust thelevel of EFL to a small positive value, preferably about 0.10±0.05%.

In the first analytical model an analysis was accomplished on aisotropic thick-walled cylinder under tensile circumferential stress.Derivation of the equations of equilibrium in the stress form weredeveloped which were similar to those disclosed in S. P. Timoshenko andJ. N. Goodier, Theory of Elasticity, 3 rd edition, McGraw-Hill, 1970.Originally published in 1934. p 65-69, which is incorporated herein byreference. Computation results were presented for 10 and 50 layers ofthe tube showing the stress distribution in the roll. Equations relatingstresses and strains were also presented for plane stress and planestrain models. Orthotropic and isotropic materials were considered.Computation results were also presented for the strain distribution, andthe distribution of EFL.

The problem of stress distribution in a roll can be considered as anaxisymmetric problem in the polar coordinate system. A typical modelelement of a thick walled cylinder 10, shown in a polar coordinatesystem, is shown in FIGS. 4A, 4B and 4C. In these Figures a element “E”of a thick-walled cylinder 10 is shown, where the element E has sides11, 12, 13 and 14.

Timoshenko (1970) presented equations of equilibrium in the polar systemof coordinates based on the equilibrium of a small element. This elementis cut out of a ring or a cylinder by the radial sections normal to theplane of FIG. 4A and is shown by bold lines 11, 12, 13 and 14, in FIG.4B. The normal stress in the circumferential direction is denoted by σ₇₃, and the stress in the radial direction as σ_(r). Components of shearstress are denoted by τ_(rΘ). The radial force on the right side 11 ofthe element is equal to (σ_(r) r dΘ) and radial force on the left side13 of the element is equal to (−σ_(r) r dΘ). The normal forces on theupper and lower perpendicular sides 14 and 12 are correspondingly(−σ_(Θ)dr dΘ/2) and (σ_(Θ)dr dΘ/2). The shearing forces on the upper andlower sides are [(τ_(rΘ))₁₂−(τ_(rΘ))₁₄]dr.

Summing up forces in the radial direction, including the body force Rper unit volume in the radial direction, produces the followingequation:

(σ_(r) r)₁₁ dΘ−(σ_(r) r)₁₃ dΘ−(σ_(Θ))₁₂ dr dΘ/2−(σ_(Θ))₁₄ drdΘ/2+[(τ_(rΘ))₁₂−(τ_(rΘ))₁₄ ]dr+R r dr dΘ=0;

Dividing by (dr dΘ) results in:${\frac{\left( {\sigma_{r}r} \right)_{11} - \left( {\sigma_{r}r} \right)_{13}}{r} - {\frac{1}{2}\left\lbrack {\left( \sigma_{\Theta} \right)_{12} + \left( \sigma_{\Theta} \right)_{14}} \right\rbrack} + \frac{\left( \tau_{r\quad \Theta} \right)_{12} - \left( \tau_{r\quad \Theta} \right)_{14}}{d\quad \Theta} + {R\quad r}} = 0.$

When the element dimensions approach infinitesimally small values, thefirst and third term of this equation represent the first derivatives,while the second term is an average value of σ_(Θ) The equation ofequilibrium in the tangential direction can be derived in the samemanner. The two equations take the following final form: $\begin{matrix}{{\frac{\partial\sigma_{r}}{\partial r} + {\frac{1}{r}\frac{\partial\tau_{r\quad \Theta}}{\partial\Theta}} + \frac{\sigma_{r} - \sigma_{\Theta}}{r} + R} = 0} & (3.1) \\{{{{\frac{1}{r}\frac{\partial\sigma_{\quad \Theta}}{\partial\Theta}} + \frac{\partial\tau_{r\quad \Theta}}{\partial r} + \frac{2\tau_{r\quad \Theta}}{r} + S} = 0},} & (3.2)\end{matrix}$

where S is the component of body force (per unit volume) in thetangential direction. When the body forces are equal to zero, equations3.1 and 3.2 can be solved using the stress function, Φ, that generallydepends on radial coordinate, r, and angular coordinate, Θ$\begin{matrix}{{\sigma_{\Theta} = \frac{\partial^{2}\Phi}{\partial r^{2}}};} & (3.3) \\{\sigma_{r} = {{\frac{1}{r}\frac{\partial\Phi}{\partial r}} + {\frac{1}{r^{2}}\frac{\partial^{2}\Phi}{\partial{\theta \quad}^{2}}}}} & (3.4)\end{matrix}$

$\begin{matrix}{\tau_{\theta \quad r} = {{\frac{1}{r}\frac{\partial\Phi}{\partial\theta}} - {\frac{1}{r}{\frac{\partial^{2}\Phi}{{{\partial r}{\partial\theta}}\quad}.}}}} & (3.5)\end{matrix}$

The stress distribution in a roll can be considered as a function ofradius only. Applying the condition of independence of the stressfunction from angular coordinate, Θ, Equations 3.3-3.5 will change asfollows: $\begin{matrix}{{\sigma_{\theta} = \frac{\partial^{2}\Phi}{\partial r^{2}}};} & (3.6) \\{{\sigma_{r} = {\frac{1}{r}\frac{\partial\Phi}{\partial r}}};} & (3.7) \\{\tau_{\theta \quad r} = 0.} & (3.8)\end{matrix}$

The form of stress function for this type of problem is suggested inTimoshenko (1970) to be:

Φ=A Ln r+B r ² Ln r+C r ² +D,  (3.9)

where A, B, and C are unknown constants.

Substitution of Equation 3.9 into Equations 3.6 and 3.7 results in thefollowing expressions for the stresses: $\begin{matrix}{{\sigma_{\theta} = {{- \frac{A}{r^{2}}} + {B\left( {3 + {2\quad L\quad n\quad r}} \right)} + {2C}}};} & (3.10) \\{\sigma_{r} = {\frac{A}{r^{2}} + {B\left( {1 + {2\quad L\quad n\quad r}} \right)} + {2{C.}}}} & (3.11)\end{matrix}$

In order to model stress conditions typical for reeled buffer tubes, theboundary conditions shown in FIG. 4C were considered. Three unknownconstants can be found from the following three boundary conditions:

1. The first layer of tape was stretched at the constant wrappingstress, σ_(w), and “glued” to the rigid core, meaning that duringdeformation the radius of the first layer will not change.

σ_(θ)(r=r _(o))=σ_(w);

2. The outer layer is a layer of tape that is stretched at the constantwrapping stress, σ_(w), and remains in the same stretched state,

σ_(θ)(r=R)=σ_(w);

3. The outer surface of the tape has no additional normal or radial load(condition of free surface):

σ_(r)(r=R)=0.

The first boundary condition corresponds to the cases of glued tape anda rigid core. Compared to experiments, this boundary condition producesa higher level of circumferential stresses in the first layer. Thesecond boundary condition represents the case of a stretched and fixedend. In the case when the upper layer is released after wrapping, whichis typical for experiments, circumferential stresses on the outer layerdrop to zero.

Application of the three boundary conditions to the system of Equations3.10 and 3.11 results in the following solutions for the unknownconstants:${A = \frac{R^{2}r_{o}^{2}\sigma_{W}L\quad n\frac{r_{o}}{R}}{R^{2} - r_{o}^{2} - {2r_{o}^{2}\quad L\quad n\frac{r_{o}}{R}}}};$${B = \frac{\left( {R^{2} - r_{o}^{2}} \right)\sigma_{W}}{2\left( {R^{2} - r_{o}^{2} - {2r_{o}^{2}\quad L\quad n\frac{r_{o}}{R}}} \right)}};$$C = {{- \sigma_{W}}{\frac{{\left( {R^{2} - r_{o}^{2}} \right)\left\lbrack {1 + {2L\quad {n(R)}}} \right\rbrack} + {2r_{o}^{2}L\quad n\frac{r_{o}}{R}}}{4\left( {R^{2} - r_{o}^{2} - {2r_{o}^{2}\quad L\quad n\frac{r_{o}}{R}}} \right)}.}}$

and subsequently the two stress components: $\begin{matrix}{{\sigma_{\theta} = {\sigma_{W}\frac{{\left( {R^{2} - r_{o}^{2}} \right)\left( {1 + {L\quad n\frac{r}{R}}} \right)} - {\left( {1 + \frac{R^{2}}{r^{2}}} \right)r_{o}^{2}L\quad n\frac{r_{o}}{R}}}{R^{2} - r_{o}^{2} - {2r_{o}^{2}\quad L\quad n\frac{r_{o}}{R}}}}};} & (3.12) \\{\sigma_{r} = {\sigma_{W}{\frac{{\left( {R^{2} - r_{o}^{2}} \right)L\quad n\frac{r}{R}} - {\left( {1 - \frac{R^{2}}{r^{2}}} \right)r_{o}^{2}L\quad n\frac{r_{o}}{R}}}{R^{2} - r_{o}^{2} - {2r_{o}^{2}\quad L\quad n\frac{r_{o}}{R}}}.}}} & (3.13)\end{matrix}$

As can be seen from Equations 3.12 and 3.13, stresses in the roll ofmaterial are linearly proportional to the wrapping stress. Thesestresses are a combination of logarithmic functions with respect to theroll radius, a hyperbolic function of 1/r², and constants representingthe core radius and outer radius of the roll.

Equations 3.12 and 3.13 were then employed to compute the stressdistribution in the roll of tape wrapped at a constant tensile stress(or draw load) around a rigid core. The example below includes thefollowing parameters: core radius 120 mm, outer radius of the roll 151.5mm, representing 10 wraps of a 3 mm thick tape, and a constant wrappingstress of 1.38×10⁵ kg/(mm s²). The level of wrapping stress was chosento be high in order to compare the solution with results of a finiteelement analysis. Young's modulus of the tape material is 1.637×10⁶kg/(mm s²) and Poisson's ratio is 0.4.

FIGS. 5A-1, -2 and -3 show the distribution of circumferential andradial stresses along the radius of the roll. The amplitude ofcircumferential stress (FIG. 5A-1) is higher than that of radial stress(FIG. 5B-2). The distribution of circumferential stress along the rollradius resembles a skewed parabola with the left and right ends at thevalues of wrapping stress prescribed as boundary conditions. When thefirst wrap is “glued” to a softer core, a certain reduction incircumferential stress in the first layers would occur. In the middle ofthe roll, reduction in circumferential stress occurs due to asuperposition of initial tensile stress and compression from upperlayers resulting in the shrinkage (radius reduction) of the layers.Similarly, if the upper layer is not held in the stretched state butreleased, the stress on the right end of the curve would also reduce. Asa result, both ends of the tape would have lesser stress than theinitial value, and the ends of the parabolic curve would move downproducing a shallower curve. This outcome would be desirable in thefabrication process as a smaller variation in the stress curves resultin lesser variations in the strains and most important, in EFL.

As can be seen from FIG. 5A-1, the “parabolic” circumferential stresscurve for the 10-layer roll has depth of 0.72%, which is the differencebetween the maximum and minimum. For 50 layers, the depth of thecircumferential stress parabola is 12.5%.

Comparison of the stress curves for 10 and 50 wraps indicated thefollowing. For a smaller number of layers, the level of radial stress issmaller than that of circumferential stress. With the increase in thenumber of layers, the role of radial forces increases.

Calculations were also performed to evaluate changes in thecircumferential stress curves when the radius of the reeled material issignificantly increased. The calculations employed the first analyticalmodel with constant core radius of 120 mm and wrapping stress of 138MPa. Comparison of these curves (shown in FIG. 5B-1 and -2) show that anincreased number of layers produced stress curves with deeper minimums.The compressive stress for 460 wraps reached 20 MPa (Ex. 1), for 2500wraps it was 220 MPa (Ex. 2), and for 5000 wraps (corresponds to a largeroll with radius of 15 m) it was 300 MPa (Ex. 3). Additional computationwith 50,000 wraps (corresponds to a roll with a very large radius of 150m) showed a level of compressive stress of about 600 MPa (FIG. 5B-2).

These results suggest that during the reeling, the depth of thecircumferential stress curve rapidly increases within the first 200-300wraps. After the first 300 wraps, subsequent growth of thecircumferential stress curve slows down (see FIGS. 5B-1 and -2). Thisindicates that the stress gradient and consequently rapid changes in EFLin the initial layers are higher than that in the outer layers.

From a practical point of view, this implies that very long buffer tubes(on the order of 1000 layers of tube per spool) can not be reeled usingconstant take-up tension on relatively small spools (core radius about100 mm) without creating high stress gradients. These stress gradientswould cause rapid changes in the EFL within the tube in a zone near thespool core that is approximately ⅓ of the total roll thickness. Thisimplies that the roll should be multi-leveled with rigid interlayers tominimize the stress compounding effect. Also, this implies that specialattention should be paid to the boundary conditions or physicalcharacteristics at the reel core surface, which includes the stiffnessof the reel core and the stiffness of compliant materials or soft padson the core surface used in the present invention. In addition, theconstant take-up tension should be replaced with a variable take-uptension, as contemplated by the present invention.

The next step taken was to equate the relationship between the stressesexperienced by the roll with strains, which directly affect the EFL of abuffer tube. In order to compute the distribution of strains in theroll, equations are needed to relate stresses and strains throughmaterial parameters such as Young's modulus and Poisson's ratio.Typically, two major models are used; plane stress or plane strain.

Generally, for elastic orthotropic materials, the relationship betweenstrains and stresses can be presented as follows: $\begin{matrix}{{ɛ_{\theta} = {\frac{\sigma_{\theta}}{E_{\theta}} - \frac{v_{\theta \quad r}\sigma_{r}}{E_{r}} - \frac{v_{Z\quad \theta}\sigma_{Z}}{E_{Z}}}};} & (3.14) \\{{ɛ_{r} = {\frac{\sigma_{r}}{E_{r}} - \frac{v_{r\quad \theta}\sigma_{\theta}}{E_{\theta}} - \frac{v_{z\quad r}\sigma_{z}}{E_{Z}}}};} & (3.15) \\{{ɛ_{z} = {\frac{\sigma_{z}}{E_{z}} - \frac{v_{\theta \quad z}\sigma_{r}}{E_{\theta}} - \frac{v_{r\quad z}\sigma_{r}}{E_{r}}}};} & (3.16) \\{{{\tau_{\theta \quad r} = \frac{\tau_{\theta \quad z}}{G_{\theta \quad z}}};{\tau_{r\quad z} = \frac{\tau_{r\quad z}}{G_{r\quad z}}};{\tau_{z\quad r} = \frac{\tau_{z\quad r}}{G_{z\quad r}}}},} & (3.17)\end{matrix}$

where ε₇₄ , ε_(r), and ε_(z) are strain components in thecircumferential (tangential), radial, and normal to the roll crosssection directions, respectively, and τ_(rθ), τ_(θz), and τ_(rz) arecomponents of shear strains.

In the plane stress model, component σ_(z)=0 (in the directionperpendicular to the roll cross section or along the axis of winding).When shear strains are negligibly small compared to the normal strains,the system of Equations 3.14-3.17 can be reduced to the following twoequations: $\begin{matrix}{{{ɛ_{\theta} = {\frac{\sigma_{\theta}}{E_{\theta}} - \frac{v_{\theta \quad r}\sigma_{r}}{E_{r}}}};}{ɛ_{r} = {\frac{\sigma_{r}}{E_{r}} - {\frac{v_{r\quad \theta}\sigma_{\theta}}{E_{\theta}}.}}}} & (3.18)\end{matrix}$

In the plane strain model, component ε_(z)=0 (in the directionperpendicular to the roll cross section or along the axis of winding).For an isotropic material the system of equations for plane strain canbe presented in the following form: $\begin{matrix}{\begin{Bmatrix}ɛ_{\theta} \\ɛ_{r} \\ɛ_{z}\end{Bmatrix} = {\begin{bmatrix}\frac{1 - v^{2}}{E} & \frac{- {v\left( {1 + v} \right)}}{E} & 0 \\\frac{- {v\left( {1 + v} \right)}}{E} & \frac{1 - v^{2}}{E} & 0 \\0 & 0 & 0\end{bmatrix}{\begin{Bmatrix}\sigma_{\theta} \\\sigma_{r} \\\sigma_{z}\end{Bmatrix}.}}} & (3.19)\end{matrix}$

When shear strains are negligibly small compared to the normal strains,the system of Equations 3.19 can be reduced to the following twoequations: $\begin{matrix}{ɛ_{\theta} = {{\frac{1 - v^{2}}{E}\sigma_{\theta}} - {\frac{v\left( {1 + v} \right)}{E}\sigma_{r}}}} & (3.20) \\{ɛ_{r} = {{\frac{1 - v^{2}}{E}\sigma_{r}} - {\frac{v\left( {1 + v} \right)}{E}{\sigma_{\theta}.}}}} & (3.21)\end{matrix}$

As can be seen from Equations 3.20 and 3.21, Poisson's ratio plays arole of stress coupling. For low values of Poisson's ratio, v→0,coupling becomes weak and$\left. ɛ_{\theta}\rightarrow\frac{\sigma_{\theta}}{E} \right.$

and $\left. ɛ_{r}\rightarrow\frac{\sigma_{r}}{E} \right.,$

which can be the case for buffer tubes.

The computations of strains and EFL in the roll using the firstanalytical model with the plane stress model was accomplished. In thisset of computations, Young's modulus of the tape material was 1.637×10³MPa and Poisson's ratio was 0.4. FIGS. 6-1, -2 and -3 depict thecomputed distribution of strains in the 10-layer roll of tape, whichcorrespond to the stresses shown in FIGS. 5A-1, -2 and -3.

It can be seen that for 10 wraps both circumferential and radial strainsare relatively small and circumferential strain monotonically decreaseswith an increase in radius (FIG. 6-1). For 50 wraps, the trend issimilar, however, the level of strain is higher than that for 10 wraps.Also, the role of radial strain has increased compared tocircumferential strain, especially in the first few layers (FIG. 6-2).

Additional computation revealed a parabolic upturn in thecircumferential strain curve for 460 wraps (FIG. 6-3). Thecircumferential strain curve shows a monotonic increase in a majorportion of the curve after approximately 100 layers. In the first fewlayers, the amplitude of radial strain is above the level ofcircumferential strain.

Assuming that the final distribution of EFL along the roll radius is adifference between the initial constant EFL_(o) value before reeling,and the circumferential strain induced by reeling, one can obtain theEFL distribution shown in FIGS. 7-1 and -2. In these calculations, theinitial constant value of EFL_(o) was taken to be 0.1.

As can be seen from FIGS. 7-1 and -2 at a constant take up stress (i.e.draw tension), 10 layers produce a relatively small variation in the EFLdistribution (Ex. 1). This variation significantly increases for 460layers (Ex. 2). In both cases, the most dramatic change in EFL (curveEx. 1) takes place in the zone close to the spool core. Computation for2500 layers revealed enhanced sharpness in the EFL curvature at thespool core surface (FIG. 7-2).

To further investigate the relationship of stress and strain on EFL ofreeled or wrapped buffer tubes, analysis under a second analytical modelwas also conducted. In this modeling variable wrapping stress andrelative stiffness of the core of the reel and the wrapped material wereanalyzed to determine a method to obtain a relatively constant EFLthroughout the length of a reeled buffer tube. In this analysis, anexisting model was modified for constant and monotonically decayingtake-up tension.

In Wolfermann, W. and Schröder D. (1987), “Web Forces and InternalTensions for the Winding of an Elastic Web,” International Conf.“Winding Technology 1987”, Stockholm, Sweden, 1987, S. 25-37, which isincorporated herein by reference, the stress distribution in rolledmaterials based on the model of a circular ring that is shrunk by awinding tension, σ_(w), was analyzed. The influence of a controlledvariation of winding tension on the stress distribution was alsoinvestigated. For anisotropic materials, equations for circumferentialand radial stresses were presented in the following form:$\begin{matrix}{\sigma_{\theta} = {\sigma_{W} - \left\lbrack {{\delta + {{{\gamma\beta}\left( \frac{r}{r_{o}} \right)}^{2\kappa}\Delta \quad \sigma}};} \right.}} & (4.1) \\{{\sigma_{r} = {{\left\lbrack {{\beta \left( \frac{r}{r_{o}} \right)}^{2\kappa} - 1} \right\rbrack {{\Delta\sigma}.{where}}\quad {\Delta\sigma}} = {\frac{1}{r^{\delta + 1}}{\int_{r_{o}}^{R}{\frac{\sigma_{W}r^{\delta}}{{\beta \left( \frac{r}{r_{o}} \right)}^{2k} - 1}{r}}}}}},} & (4.2)\end{matrix}$

parameters γ, δ and κ represent anisotropic properties of the wrappedmaterial, and β is the parameter relating stiffness of the wrappedmaterial to that of the core.

When Young's modulus of the core material is much higher than that ofthe wrapped material, β≈−1, and can be as low as −2 for very stiff corematerials. This range of values for β was considered by Wolferman andSchroder to model the stress distribution in paper rolled on a steelcore. For the case of paper rolled on a paper roll, the authorssuggested β=2. According to Wolferman and Schroder, radially shrinkinglayers influence the circumferential stresses in the middle part of theroll resulting in compressive circumferential stresses, with the largestcompressive stresses achieved when β=2. Thus, it was recommended to usecores made of hard materials. Also, the authors recommended use of atwo-stage function for wrapping stress to reduce the range of the stressvariation. Initially, wrapping should be performed under a constant highlevel of wrapping stress. After a certain number of wraps, the wrappingstress should be monotonically reduced as shown in the figure below:

Two-stage Variable Take-up Tension from Wolfermann and Schroder.

For isotropic materials, γ=δ=κ=1 and Equations 4.1 and 4.2 can besimplified to $\begin{matrix}{{\sigma_{\theta} = {\sigma_{W} - {\frac{r_{o}^{2} + {\beta \quad r^{2}}}{r^{2}}{\int_{r_{o}}^{R}{\frac{\sigma_{W}r}{{\beta \quad r^{2}} - r_{o}^{2}}{r}}}}}},} & (4.3) \\{\sigma_{r} = {\frac{r_{o}^{2} - {\beta \quad r^{2}}}{r^{2}}{\int_{r_{o}}^{R}{\frac{\sigma_{W}r}{{\beta \quad r^{2}} - r_{o}^{2}}{{r}.}}}}} & (4.4)\end{matrix}$

Several candidate functions were considered, mainly from a family ofsmooth functions for the radius-dependent take-up stress σ_(w). Linearlydecaying functions for σ_(w) produced complicated integral expressionsfor which closed-form solutions were not obtained. In contrast,monotonically decaying take-up stress in the parabolic form produced arelatively simple integral expression. This type of parabolic functionis shown below: $\begin{matrix}{{\sigma_{w} = {\sigma_{wo}\left( {1 - \frac{\alpha \quad r^{2}}{2R^{2}}} \right)}},} & (4.5)\end{matrix}$

where the parameter σ_(wo) is the initial value of take-up stress, and adefines the decay rate; α=0 for constant take-up tension, small valuesof α for slow decay, and larger values of α for rapid decay. FIG. 8depicts these three cases of decay rate. It is noted that monotonicallydecaying take-up tension can be achieved by, or is substantiallyequivalent to, using a bucket of water with a valve for the slow releaseof the water over time, thus decreasing the tension load over time.

For the family of take-up stress functions with parabolic decay,circumferential and radial stress components were found by substitutionof Equation 4.5 into Equations 4.3 and 4.4: $\begin{matrix}{{\sigma_{\theta} = {\sigma_{wo}\left\{ {1 - \frac{\alpha \quad r^{2}}{2R^{2}} + {\frac{r_{o}^{2} - {\beta \quad r^{2}}}{\left( {2\beta \quad {rR}} \right)^{2}}\left\lbrack {{{\alpha\beta}\left( {r^{2} - R^{2}} \right)} + {\left( {{2\beta \quad R^{2}} - {\alpha \quad r_{o}^{2}}} \right){Ln}\quad \frac{{\beta \quad R^{2}} - r_{o}^{2}}{{\beta \quad r^{2}} - r_{o}^{2}}}} \right\rbrack}} \right\}}},} & (4.5) \\{\sigma_{r} = {\sigma_{wo}{{\frac{r_{o}^{2} - {\beta \quad r^{2}}}{\left( {2\beta \quad {rR}} \right)^{2}}\left\lbrack {{{\alpha\beta}\left( {r^{2} - R^{2}} \right)} + {\left( {{2\beta \quad R^{2}} - {\alpha \quad r_{o}^{2}}} \right){Ln}\quad \frac{{\beta \quad R^{2}} - r_{o}^{2}}{{\beta \quad r^{2}} - r_{o}^{2}}}} \right\rbrack}.}}} & (4.6)\end{matrix}$

In the case of constant take-up tensile stress, when α=0,$\sigma_{\theta} = {\sigma_{wo}\frac{{3\beta \quad r^{2}} - r_{o}^{2}}{2\beta \quad r^{2}}{Ln}\quad \frac{{\beta \quad R^{2}} - r_{o}^{2}}{{\beta \quad r^{2}} - r_{o}^{2}}}$

and $\begin{matrix}{\sigma_{r} = {\sigma_{wo}\frac{r_{o}^{2} - {\beta \quad r^{2}}}{2\beta \quad r^{2}}{Ln}\quad {\frac{{\beta \quad R^{2}} - r_{o}^{2}}{{\beta \quad r^{2}} - r_{o}^{2}}.}}} & (4.7)\end{matrix}$

Computations were made using the Mathematica® software and involvedcomplex numbers,$\left. {{Ln}\frac{\quad {{\beta \quad R^{2}} - r_{o}^{2}}}{{\beta \quad r^{2}} - r_{o}^{2}}}\rightarrow \right. = {A + {iB}}$

and stresses appeared to be expressed as follows:{σ_(θ),σ_(r)}=Function{D+iG}, where D and G are constants, and G is anegligibly small number. Thus, final expressions for stress componentscontained real numbers only.

Equations 4.7 were applied to the following case: core radius is 120 mm,outer radius of the roll is 151.5 mm, and constant wrapping stress is1.38×10⁵ kg/(mm S²). Young's modulus of the tape material is 1.637×10⁶kg/(mm s²) and Poisson's ratio is 0.4. FIGS. 9A-1, -2 and -3 show thedistribution of the stresses in the roll obtained by using the secondmodel with a rigid core; i.e. β=−1.

A comparison of the stress distribution computed using the firstanalytical model (a cylinder with tensile stresses) and the secondanalytical model (a shrunk ring) is shown in FIGS. 9B-1 and -2 for 10(FIG. 9B-1) and 2460 layers (FIG. 9B-2). Curves for radial stressesalmost coincide while the curves for circumferential stresses are veryclose to each other. The first model produced a curve with a parabolicshape and is located up to 0.7% above that obtained by using the secondanalytical model with β=−1. Additional computation for β=−2, whichrepresents a very rigid core, showed a difference in circumferentialstresses up to 4.4% between the first and the second models with themaximum difference occurring near the core at r=r_(o). A smalldifference in the radial stresses was also found at the beginning of theroll.

FIGS. 9C-1 and -2 show a comparison of the strain distribution obtainedfrom the first and second models for 10 (FIG. 9C-1) and 2460 (FIG. 9C-2)layers of material using plane stress equations. As can be seen thecurves obtained from the two models are very similar.

In conducting the above research it was needed to identify EFL as afunction of several main parameters representing geometry, materials,and processing. During the research it was found that EFL is sensitiveto the following parameters and factors:

Roll geometry—initial and final radius, r_(o)<r<R;

Stiffness of the reel core compared to that of buffer tube, β;

Young's modulus representing buffer tube material that depends on timeand temperature, E(t, T);

Poisson's ratio, v;

Initial level of EFL, before reeling; and

Take-up stress function, including amplitude and decay rate α.

The EFL on the reel can be computed based on the value of EFL_(o) beforereeling and strain in circumferential direction as:

EFL=EFL _(o)−ε_(θ)  (4.8)

that is: $\begin{matrix}{{{EFL} = {{EFL}_{o} - {\frac{\sigma_{wo}}{E\left( {t,T^{o}} \right)}\frac{\left( {1 + v} \right)}{4}\left\{ {{\left\lbrack {\left( {1 - {2v}} \right) + \frac{r_{o}^{2}}{\beta \quad r^{2}}} \right\rbrack \left( {\frac{\alpha \quad r_{o}^{2}}{\beta \quad R^{2}} - 2} \right){Ln}\quad {\frac{{\beta \quad R^{2}} - r_{o}^{2}}{{\beta \quad r^{2}} - r_{o}^{2}}++}\left( {{4v} - 3} \right)\frac{\alpha \quad r^{2}}{R^{2}}} + {\left( {\frac{1}{r^{2}} - \frac{1}{R^{2}}} \right)\frac{\alpha \quad r_{o}^{2}}{\beta}} + {\alpha \left( {1 - {2v}} \right)} - {4\left( {1 - v} \right)}} \right\}}}}{or}} & (4.9) \\{{EFL} = {{EFL}_{o} + \frac{\sigma_{wo}\left( {1 - v^{2}} \right)}{E\left( {t,T^{o}} \right)} - {\frac{\sigma_{wo}}{E\left( {t,T^{o}} \right)}\frac{\left( {1 + v} \right)}{4}{\left\{ {{\left\lbrack {\left( {1 - {2v}} \right) + \frac{r_{o}^{2}}{\beta \quad r^{2}}} \right\rbrack \left( {\frac{\alpha \quad r_{o}^{2}}{\beta \quad R^{2}} - 2} \right){Ln}\quad {\frac{{\beta \quad R^{2}} - r_{o}^{2}}{{\beta \quad r^{2}} - r_{o}^{2}}++}\left( {{4v} - 3} \right)\frac{\alpha \quad r^{2}}{R^{2}}} + {\left( {\frac{1}{r^{2}} - \frac{1}{R^{2}}} \right)\frac{\alpha_{o}^{2}}{\beta}} + {\alpha \left( {1 - {2v}} \right)}} \right\}.}}}} & (4.10)\end{matrix}$

As can be seen from equations 4.8 and 4.9, EFL can be considered as adifference between the initial EFL (before reeling) and a strain on thereel. Also, the strain on the reel is a product of the initial tensilestrain (initial stress divided by time- and temperature-dependentmodulus of elasticity), shown as the term underlined in a solid line,and a reeling function, shown as the term underlined in a dashed line.These quantities depend on the Young's modulus, Poisson's ratio,relative stiffness of the buffer tube and reel core, and decay rate inthe take-up tension. Poisson's ratio represents the degree of couplingbetween the circumferential and radial stresses. The expression for EFLcan be rearranged, according to Equation 4.10, to show the contributionfrom the bending stiffness per unit area or unit thickness, E/(1-v²),and the stretching stiffness that is proportional to Young's modulus.

Several experiments revealed that when subjected to constant tension,thermoplastic materials exhibit a monotonic reduction in the Young'smodulus apparently due to reorientation of molecular chains. Forthermoplastic buffer tube materials, the reduction in the Young'smodulus was as large as two times. Numerical computations performedusing Mathematica®, and Equation 4.10 revealed that the depth of the EFL“parabola” increases with a decrease in Young's modulus.

The time factor and long-term stretching of buffer tubes influence notonly Young's modulus, but also creep and shrinkage behavior of thetubes. For high line speeds, buffer tubes are subjected to tension(before the reeling stage) for a shorter duration. Consequently,elongation of thermoplastic materials are smaller than that for lowerline speeds. As a result, the EFL obtained for high-speed lines areabove that obtained for lower-speed lines.

Typically, an increase in the line speed is associated with a shortertime to cool the buffer tube in the cooling system 381 (because the tubeT is passing through a cooling apparatus 381 quicker). See FIG. 38.Consequently there is an increased temperature of the reeled buffer tubeafter manufacture that in turn results in residual shrinkage in additionto that caused by the other residual forces experience while on the reeland during reeling. However, elongation of the reeled material underexisting residual tension (circumferential stress) is generally moresignificant than the shrinkage of thermoplastic materials while cooling.The action of circumferential stresses can be related to the creep ofthermoplastic materials. Reeled material under circumferential stresscan undergo a certain amount of elongation, thus reducing the level ofstresses. To some extent, a longer amount of time on the reel produceslower EFL. Consequently, creep as a function of the time spent on thereel can also be used with the present invention to adjust (reduce) EFLto the desired level.

Also, as discovered in a second embodiment of the present invention,variations in the line speed and the corresponding variation in angularvelocity (“ω”) of the reel produce a variation in temperature andtensile load through the radius of the buffer tube roll. In thisembodiment, a monotonically variable angular velocity of the spool isused to control the stress state in the buffer tubes, and subsequentlythe EFL distribution. It should be noted that it is preferred to use themonotonically variable angular velocity of the spool of this embodimentwith a stiffness-compliant pad on the reel core to achieve asubstantially even EFL distribution. This will be discussed in moredetail below.

Examples of two possible monotonically variable curves for the angularspeed of the spool are shown in FIG. 10. The curves shown 103 and 104represent two out of many possible monotonically variable curves forangular velocity. It is noted that the exact shape of the monotonicallyvariable curves will depend on several factors including materialproperties of the reeled material and spool, additional variation intake-up tension, and stiffness of pad(s) used, and is to be adjusted foreach individual manufacturing line. For a typical case of a regularrigid (steel) spool core, the curve 104 in FIG. 10 is proposed (but anysimilar curve may be used), while if a pad is used on the spool thecurve 103 is proposed. Initially, the angular velocity of the spoolmonotonically increases. This increase produces monotonically increasedtake-up tension. As a result, the buffer tube elongates, from a smalllevel at the beginning of the tube to a higher level, causingcorresponding changes in the EFL levels, from higher to lower. Becauseof this, the left side of initial EFL parabola (dashed curverepresenting an EFL for constant velocity spooling) turns down as shownby arrow 101.

Further, in the preferred embodiment, the ramping rate of angularvelocity of the reel, or spool, slows down to produce smaller take-uptension and to increase the EFL. This is shown in FIG. 10 by arrow 102.This slow-down step is especially important for the middle part of theroll. Continuous slow down to the end of the buffer tube will provide amonotonically reduced tension and a flattened EFL curve as shown in FIG.10 by arrow 113 which indicates flattening of the EFL curve.

It is anticipated that some deformations of thermoplastic materials arereversible (elastic) while the others are permanent (plastic). That iswhy the direct comparison of the EFL in the reeled material remaining onthe spool to that unreeled is not always accurate. The EFL after reelingcan be computed based on the value of EFL before reeling and plastic orresidual strain, ε_(θ) ⁹² , in the circumferential direction:

EFL _(Final)=EFL−ε_(θ) ⁹² .  (4.11)

The changes in microstructural properties and associated elongationsleading to changes in EFL are summarized in FIG. 11, which shows arelationship between tensile stress, shrinkage, time on reel and creepand how they may increase EFL of a buffer tube.

In the present invention, an expression for the decay parameter can bederived from Equation 4.10. As a practical approach, several cases withvariable α should be considered to determine through iteration when theEFL is close to a desired constant level. The exact application of thisformula would vary with relation to manufacturing facility andparameters, and should be optimized for each individual calculation toensure that the desired level of EFL is maintained.

An example of this is demonstrated in FIG. 12. In this Figure, a numberof examples are shown, where the tension decay rate was varied for fourtypical cases of reel core stiffness. Odd numbered curves in FIG. 12correspond to circumferential stress. The goal was to obtain aconstant-value distribution of circumferential stress. Curves 121, 123,125, 127, 129 and 1211 show a sequence of the variable core stiffness,from “regular” rigid (121 and 123) to soft core (125 and 127), then to arigid core with a thin soft pad (curve 129) and finally, a core withincreased stiffness (curve 1211). Numerical experiments revealed thatthe distribution of circumferential stress is very sensitive to the corestiffness and the decay rate in tension. The even number curvesrepresent radial stresses as compared to roll radius.

FIG. 13 presents a wider variety of computed cases with differentrelative core stiffness and decay rates of take-up tension. The numbersin the Figure represent minimum and maximum levels of circumferentialstress. The shapes of the curves in FIG. 13 show periodicity in thestress distribution and suggest how to control the stiffness of the coreand take-up tension in order to achieve constant circumferential stressin the roll and subsequently, constant values of EFL. In particular,computations for α=0.6 and β=−1.2 produce a variation in thecircumferential stress in the range from 92 to 96 MPa; i.e. within±2.5%.

Analysis of the distribution of circumferential stress suggests thatthere are three major zones. The first one is a zone of unstablesolutions (or unstable behavior) in the stress distribution when theparameter β is positive. For different rates of decay in take-up load,the curves show a sharp transition to lower (compressive) stresses atthe core surface. This may or may not be attributed to the properties ofthe logarithmic functions alone. The second zone corresponds to negativevalues of the parameter β, in the range from −1 to 0. In most cases, theregion near the core surface exhibits a higher level of circumferentialstresses. The curves for circumferential stress decay from the point onthe reel core to the outer layer, often forming a minimum in the middlepart of the roll. Variation between the maximum and minimum values inthis zone is smaller than that in the first zone. Also, this variationseems to be minimal for β close to −0.8 and small values of α (gray zonein the table). The third zone corresponds to a spool core with increasedstiffness; i.e. β=−2. Several stress curve shapes are possible in thiszone, generally with a small variation between the minimum and maximumvalues, with smaller variation more typical for a rapid decay in take-uptension.

Another validation analysis performed in the research and development ofthe present invention was a finite element analysis of a buffer tubewound on a reel. This was done because the stress distribution along thelength of a buffer tube has an influence on the excess fiber lengthwithin the tube. A variation of stresses exists in the wound structurethat results from the combined loading state of the applied longitudinaltension and radial compression due to the interaction of multiplelayers. Several models were developed to simulate the process of windinga material under tension onto a reel. In each of the models, a widesheet of material, with some given thickness, was used to approximateone full traverse of buffer tubes on a reel. This was done in order toreduce the problem to a two-dimensional, plane strain condition, since afull three-dimensional model of a long buffer tube would not befeasible. A plane strain analysis of the problem is reasonable becausethe most significant factor influencing the stress variation is thecompression due to the multiple layers.

The first FEA model developed included a spiral-shaped sheet of materialinitially coiled around a reel in a stress free state. The outerdiameter of the reel was 200.0 mm, and the thickness of the sheet was3.0 mm. The structure consisted of ten layers L, with a gap of 0.5 mmbetween the layers to allow for tightening of the coil. The pre-coiledstate helps to eliminate the numerical difficulties associated withdynamic winding of the material around a rotating reel. The finiteelement mesh for this model is shown in FIG. 14A. The surface of thereel was created using a spiral so there would be a slight offset wherethe sheet is connected. This was done to eliminate a sharp corner at theattachment point, in order to prevent stress concentrations. Four nodedquadrilateral plane strain elements, with reduced integration were usedfor the sheet. The sheet had three elements E* through the thickness,and 3800 elements along the length. Contact surfaces were established onthe outer surface of the reel and inner surface of the sheet, andself-contact surfaces were established on the sheet to handle thecontact between overlapping layers.

The reel was modeled as a rigid body, and the sheet was modeled as anelastic material with the properties of polypropylene at roomtemperature. The elastic modulus was taken to be 1.637 GPa, Poisson'sratio was taken as 0.40, and the density was taken to be 900 kg/m³.

An explicit dynamic analysis was performed with a time-varying forceapplied to the nodes on the top surface of the outer layer of the sheet.The reel was constrained from translation and rotation, and the nodes onthe bottom surface of the inner layer of the sheet were constrained fromtranslation. A force of 10.0 N was applied to each node in the top threerows on the outer layer of the sheet. The rise time for the force was10.0 seconds, and the analysis was conducted for a total time period of20.0 seconds, in order to achieve a steady state solution. The densityof the sheet material was scaled by a factor of 10⁴ in order to achievea reasonable time step.

FIG. 14B shows the reference and deformed configurations of thestructure after 11.0 seconds. The deformed view shows the lengthincreasing as the layers come into contact, and the gap between layersdisappears.

FIG. 14C shows a plot of the stresses, σ₁₁ and σ₂₂, in the sheet at 11.0seconds. A local coordinate system was used so that σ₁₁ is along theaxis of the sheet, and σ₂₂ is through the thickness. When consideringthe wound structure, the stresses σ₁₁ and σ₂₂ correspond to stresses inthe circumferential and radial directions, respectively. The main unitsused in the analysis were kg, mm, and seconds, so the magnitude ofstresses shown in FIG. 14C needs to be multiplied by 1000 to obtainPascals. The maximum circumferential stress is observed in the outerlayer, and the value decays to zero at the inner layer. To compute thestresses for the plane strain condition, an effective width out of theplane of 1.0 mm is assumed. Using the total maximum applied force of 120N, a simple static calculation of force divided by area for a rod givesa longitudinal stress of 40.0 MPa.

The distribution of radial stress shows the maximum stress occurring inthe inner layer adjacent to the reel surface, and zero stress in theouter layer. The pattern of stress observed is reasonable for the typeof load applied, but it does not show a strong interaction between thecompressive forces and the circumferential stress. In the actual windingprocess, each layer of material added to the reel is under an appliedtension, so the stresses evolve from the inner layer outwards. Themodel, however, has the stress starting at the outer layer due to theapplied tension, and then propagating inwards towards the reel.

In order to obtain a more accurate representation of the problem, adynamic winding model was developed. This model included a reel with along sheet of material attached to the side. The outer diameter of thereel was 240.0 mm, and the thickness and length of the sheet was 3.0 and7000.0 mm, respectively. The length of the sheet was chosen to makeapproximately ten wraps around the reel. The model and finite elementmesh used are shown in FIG. 15A. The surface of the reel was createdusing a spiral so there would be a slight offset where the sheet isattached. This was done to eliminate a sharp corner at the attachmentpoint, in order to prevent any stress concentrations. The nodes on thebottom edge of the sheet were made coincident with the nodes on the reelsurface to create a perfect bond. Four noded quadrilateral plane strainelements E, with reduced integration, were used for the sheet 151. Thesheet 151 had three elements through the thickness, and 4000 elementsalong the length. Contact surfaces were established on the outer surfaceof the reel and inner surface of the sheet, and self-contact surfaceswere established on the sheet to handle the contact between overlappinglayers. The reel was modeled as a rigid body, and the sheet was modeledas an elastic material with the properties of polypropylene at roomtemperature. The elastic modulus was taken to be 1.637 GPa, Poisson'sratio was taken as 0.40, and the density was taken to be 900 kg/m³.

An explicit dynamic analysis was performed with a force applied to theend of the sheet, followed by the application of angular velocity to thereel. The reel was constrained from translation, but was allowed torotate under the action of the prescribed angular velocity.

In the computations, a force of 10.0 N was applied to the first tenhorizontal rows of nodes at the top end of the sheet. The ramp time onthe force was 2.0 seconds, and the angular velocity started at 3.0seconds and attained its steady value by 5.0 seconds. The force wasapplied before the angular velocity to allow the transients in the sheetto die out before winding occurred. An angular velocity of 9.7 rad/secwas chosen to achieve an approximate linear velocity of 70.0 m/min forthe sheet. The variation of the force and velocity in time is depictedin FIG. 15B. The density of the sheet material was scaled by a factor of10⁴ in order to achieve a reasonable time step.

FIG. 15C shows a plot of the circumferential and radial stresses in thewound sheet at 10.43 seconds. The magnitude of stress needs to bemultiplied by 1000 in order to obtain units of Pascals. This plot showsa snapshot of the stresses at the time the sheet is almost completelywound. The circumferential stress is the highest in the layer closest tothe reel surface, and it drops rapidly within the next few layers. Thestress changes from 290 MPa in the first few layers, to approximately 90MPa in the middle layers, and then to about 150 MPa in the outer layers.The plot of radial stresses shows that the inner layers near the surfaceof the reel are under compression, at about −40 MPa, and the middlelayers have a stress level of approximately −20 MPa. In the outerlayers, the radial stresses are close to zero. The plot of radial stressshows a significant amount of noise, but the trend of near zero stressin the outer layers, and increasing compressive stresses in the innerlayers is as expected. The “noisy” distribution of stresses shown by theFEA model are most likely due to the contribution from bending. Layersof the buffer tube closer to the reel are subjected to higher bendingstress gradients as compared to the outer layers.

A graph of the circumferential strain over the length of the sheet isshown in FIG. 15D. This plot shows the high spike in strain that occursnear the inner surface of the reel, and a slight variation in strainover the rest of the length. The spike in strain could be the result ofan instability that occurred during the winding process. After a fewlayers were taken up on the reel, the layers briefly loosened, and someslack was introduced into the wound structure. The slack quicklydisappeared, however, and the winding continued smoothly for the rest ofthe analysis. The addition of slack to the system, and the subsequentrecovery could have resulted in the sharp increase in the strain.

The major difficulty with the dynamic winding simulation is maintainingstability of the winding process. Simulations were performed in whichthe choice of tension and velocity values, and loading rates resulted inlayers coming completely off the reel. The solution also consumes alarge amount of computation time, so slowing down the velocity or theloading rate causes the analysis to take even longer to run.

A simulation was performed with a force rise time of 10.0 seconds, andwith the velocity starting at 15.0 seconds and attaining its steadyvalue by 25.0 seconds. The winding of the entire length of the sheetcompleted smoothly without any interruptions or instabilities. Thedistribution of circumferential and radial stress at 26.5 seconds isshown in FIG. 16. The stresses are very similar to the fast loading ratecase, except the large circumferential stress near the surface of thereel is not present. A plot of the circumferential strain, shown in FIG.17, illustrates the influence of the loading rate. The strain for theslower loading rate case is relatively smooth along the length, and doesnot have the large spike present in the higher loading rate case.Assuming that the faster loading rate case is not valid due to theinstability, the slower case would have to be used to study the effectof winding on the stress and strain distribution. The strain for theslower case is relatively flat which suggests that there are not enoughlayers to cause a significant interaction between the radial compressionand axial tension. This creates the need for a longer sheet of material,which would drastically increase computation time, and the possibilityof instability.

The computation time for the slow loading rate case was approximatelytwenty hours, compared to ten for the fast case. An analysis wasattempted on a longer sheet with a length of 12000.0 mm. The solutioncould not complete due to problems with excessive deformation after onlyabout one wrap around the reel. Assuming a suitable set of loadingconditions could be found to allow for complete winding, the solutiontime is still too large for the model to be practical.

FIG. 18 depicts the results obtained from the second analytical modeland the finite element model. As expected, due to inertia forces, fastramping produced a wider level of “noise” in the stress curve ascompared to slower ramping. Average levels of the circumferentialstresses obtained from the FEA model are close to each other for fastand slower ramping. The range of stresses agrees with the the analyticalsolutions for the cases of β=−1 and β=−2.

A third model was developed that was suitable for studying theinteraction between circumferential tension and radial compressionwithin a reasonable amount of computation time. The model consisted ofconcentric rings, wrapped around a reel, which were incrementallyactivated into the solution with an initial tensile stress. A staticequilibrium solution was obtained for each underlying layer beforeadditional layers were added. This model also used a two dimensional,plane strain assumption for approximating the interaction of layers ofbuffer tubes on a reel.

Since each layer is a complete ring, quarter symmetry was used for theproblem. A smaller section could have been modeled, but the quartersection makes the application of boundary conditions easier, and allowsthe solution to be obtained away from any end effects. The outerdiameter of the reel was 240.0 mm, and the thickness of each layer was3.0 mm. Fifty layers were modeled in order to have a significant amountof radial compression. Each layer was modeled with a single elementthrough the thickness, and 80 elements along the length of the quartersection. Four-noded quadrilateral plane strain elements, with reducedintegration, were used. The finite element mesh is shown in FIG. 19A.

The nodes on the inner surface of the first layer were constrained inorder to simulate the rigid surface of the reel. Symmetry conditionswere applied to the horizontal and vertical surfaces. All the layerswere perfectly bonded together so there were no contact surfacesdefined. The layers were modeled as a linear elastic material with anelastic modulus of 1.637 GPa, and a Poisson's ratio of 0.10. The Poissonratio was lowered from the typical value of 0.40 because it seems tohave an amplifying effect on the variation in stresses. Since buffertubes are typically hollow and gel filled, the Poisson ratio isdifferent than that of solid polypropylene material.

The analysis for the concentric layer model starts with an initialtensile stress applied to the first layer, for which an equilibriumsolution is then computed. The remaining layers are not considered inthe solution for this step. At the next step, the first layer will havesome stress state, and the second layer will be activated with theinitial stress value. The equilibrium solution will then be computed forboth layers. At the end of the first step, the nodes common to the firstand second layer may have moved, but the second layer will be activatedstrain free in the second step. The deformations are small, so the shapeof the elements did not change significantly. The analysis was continueduntil all layers have been activated, and a final equilibrium state isdetermined.

The analysis was performed with an initial stress value of 10.0 MPa, tosimulate a winding tension of 30.0 N. FIG. 19B shows the circumferentialand radial stress distributions at the final state. The stress valuesshown need to be multiplied by 1000 to obtain units of Pascals. Theradial stress plot indicates zero stress at the outer surface, and thehighest compression at the inner surface. The circumferential stressplot shows a high stress in the inner and outer layers, but a lowerstress in the interior. This stress distribution would translate into avariation in stress or strain along the length of a wound material suchas a buffer tube.

The circumferential strain, which is a function of both thecircumferential and radial stresses, can be interpreted as thecircumferential strain in each layer of wound material. In order tounderstand how the EFL would vary along the length of a buffer tube, itis necessary to look at the circumferential strain. The shape of thecircumferential strain distribution along the length can be ascertainedfrom the circumferential strain through the thickness of the layers. Ifthe concentric layers represent a wound buffer tube, the strain in eachlayer can be interpreted as a sampling of the strain along the length.The stress and strain distribution through the thickness of the layersis shown in FIG. 19C.

If the EFL in the buffer tube is constant before the tube is taken up onthe reel, it can be assumed that the circumferential strain induced bythe winding will directly affect the amount of EFL. The relation betweenstrain and percent EFL can be stated as:

EFL=EFL _(o)−100*ε  (5.1)

where ε is the circumferential strain, and EFL_(o) is the initialpercent EFL. The EFL distribution computed for the case of 30.0 Nconstant tension is shown in FIG. 19D. The circumferential strain hasbeen used to approximate the axial strain, and the length has beennormalized to one. An initial EFL of 0.6% was assumed for this case. TheEFL curve has the distinct parabolic shape that is observed in theexperiments.

The concentric layer model was used to run various cases in order toprovide a better understanding of the mechanisms influencing the straindistribution. Simulations were performed to determine the effect ofmaterial modulus on the strain. Three values of elastic modulus werechosen, 0.1637, 1.637, and 16.37 GPa. The Poisson ratio was kept at 0.1for each case, and a constant tension of 30.0 N was used. Plots of theradial and circumferential stress and strain are shown in FIG. 20. Asexpected, a higher Young's modulus decreases the circumferential strain,and flattens out the curve. This indicates that the winding processwould have less of an effect on the EFL distribution for a stiffermaterial. This is consistent with observations of a more uniform EFLdistribution in PBT buffer tubes, which have a higher modulus than thepolypropylene equivalent.

Another parameter influencing the strain distribution is the diameter ofthe reel core. Simulations were performed with core diameters of 120.0,240.0, and 480.0 mm. The total number of layers was kept the same foreach case, so the total material thickness was 150.0 mm. The materialproperties and the applied tension were also kept the same for eachcase. The modulus was taken to be 1.637 GPa, Poisson's ratio was 0.1,and the tension was constant at 30.0N. Plots of the radial andcircumferential stress and strain are shown in FIG. 21. The radius onthe x-axis was changed to start at the outer surface of the core insteadof at the center, in order to directly compare the results. The stressand strain variation is much larger for the smaller core diameter, andthe parabolic shape of the circumferential strain curve is much morepronounced. The length of material is greater for the larger corediameters, but since the thickness of the material is the same for eachcase, the influence of the bend radius can still be determined. Theobservation of a greater variation in stress and strain for the smallercore diameter is consistent with experiments conducted on two differentsized reels.

Simulations were performed to investigate the effect of the tensionlevel on the stress and strain distribution. The tension was keptconstant in each case, and the values chosen were 10.0, 20.0, 30.0,40.0, and 50.0 N. The modulus for each case was taken to be 1.637 GPa,and Poisson's ratio was 0.1. Plots of the radial and circumferentialstress and strain are shown in FIG. 22. As expected, the increase intension results in higher radial compression of the layers, and highercircumferential strain. Also, a larger variation in circumferentialstrain occurs with increasing tension, which would result in a greatervariation in the EFL.

The effect of a variable tension was investigated by changing theinitial stress for each of the layers. Simulations were performed withthe tension starting at 30.0 N, and linearly decaying to values of 25.0,20.0, 15.0 and 10.0 N, respectively. The range of tension values wasbroken into fifty increments, and the corresponding initial stresseswere assigned to the appropriate layers. Plots of the radial andcircumferential stress and strain are shown in FIG. 23. The 30.0 Nconstant tension case is also shown in the graph for comparison. Theplot shows that the decay in tension has only a slight influence on theradial stress and strain, but has a very strong effect on thedistribution of circumferential stress and strain. The appropriatechoice of tension decay will flatten out the right portion of thecircumferential strain curve, but a large variation will still existbetween the left and right sides of the curve. If the starting tensionwas increased or decreased, the variation would still be present, but itmay change in magnitude.

In the present invention, the addition of a compliant layer (such as asoft pad) to the surface of the reel could influence the straindistribution in the wound material. In order to simulate a compliantlayer on the reel, different material properties were assigned to thefirst layer in the model. The modulus for the regular material was takento be 1.637 GPa, and Poisson's ratio was 0.1. Several cases were runwith the modulus of the first layer reduced by 10.0, 50.0, and 100.0times that of the regular material. The Poisson's ratio was 0.1 for thefirst layer in each case. Additional cases were run using the same setof reduced modulus values for both the first and second layers. Thetension was kept constant at 30.0 N for each case. Plots of the radialand circumferential stress and strain are shown in FIG. 24. The curvelabeled as baseline in the graphs is the case without a compliant layer.The stress and strain for the compliant layers are not shown in thegraphs because they experience large deformation. Also, the stresslevels in the compliant layer are of no interest since the rest of thelayers would be representing the wound buffer tube material. The dataindicates that the compliant layer helps to reduce the radialcompression within the layers, with the most severe cases changing theconcavity of the radial stress and strain curves. The reduction inradial compression causes a drop in the circumferential strain withinthe layers closest to the reel surface. A large reduction in the modulusof the compliant layer causes the circumferential strain to change froma state of tension to a state of compression in the layers near thesurface of the reel. The concavity of the circumferential strain curvealso changes as the modulus of the compliant layer is reduced.

The concentric layer model was used to study the effect of compliantlayers distributed among the regular material layers on the stress andstrain distribution. A simulation was performed using a material with areduced modulus in place of the regular material for layers number ten,twenty, thirty and forty. An additional simulation was performed forcompliant material in place of layers ten, eleven, twenty, twenty one,thirty, thirty one, forty, and forty one. The modulus of the compliantmaterial was reduced by a factor of thirty from the regular material toa value of 0.5456 GPa, and Poisson's ratio was taken to be 0.1. Plots ofthe radial and circumferential stress and strain are shown in FIG. 25A.The baseline case without compliant layers is also shown in the plots.The stress and strain values within the compliant layers are notincluded in the curves because the main focus is the resultingdistribution within the regular material. The curves show that thecompliant layers cause a significant amount of variation in thecircumferential stress and strain when compared to the baseline case.This would lead to a greater variation in the EFL distribution than thebaseline case.

As an alternative to the distributed compliant layers, a case wasconsidered with distributed stiff layers. Simulations were performedwith stiff layers in the same configuration as the compliant layersdiscussed previously. The modulus of the stiff layers was increased tentimes that of the regular material, to a value of 16.37 GPa, andPoisson's ratio was taken to be 0.1. An additional case was consideredwith a modulus one hundred times greater than the regular material, or163.7 GPa. Plots of the radial and circumferential stress and strain areshown in FIG. 25B. The curves show that the stiff layers act to reducethe variation in circumferential stress and strain, and therefore wouldreduce the variation in EFL.

Another case that was considered to control the distribution in EFL wasan expandable core. This case was modeled by removing the constraints onthe boundary of the inner layer, and applying a normal pressure to thesurface. The boundary was released, and the pressure was applied afterall layers were added. Simulations were performed with pressures of10.0, 20.0, and 40.0 MPa, respectively. Plots of the radial andcircumferential stress and strain are shown in FIG. 25C. The curves showthat the pressure shifts the circumferential stress and strain curvesup, and also changes the shape of the curves. The effect of the pressureis more pronounced within the layers closest to the reel surface. Theincrease in strain in the layers resulting from the pressure would causea decrease in EFL. This effect would need to be combined with anothermethod of EFL control in order to obtain a desirable EFL distribution.If this technique were to be used, it would need to be determined if thepressure would damage the layers of buffer tube near the reel surface.The curve for the 10.0 MPa pressure case is below the baseline curve,which indicates that the pressure was not high enough, and the layersmoved inwards from the surface of the reel.

Simulations were performed to investigate the combined effect ofvariable tension and compliant layers. A model with a compliant layerwas used to run different cases of linearly decaying tension. Themodulus of the compliant layer was 0.12 GPa, and the modulus of theregular material was 1.637 GPa. The Poisson ratio for both materials was0.1. The tension was linearly decayed from a starting value of 30.0 N,to values of 25.0, 20.0, 15.0, and 10.0 N, respectively. Plots of theradial and circumferential stress and strain are shown in FIG. 26A. Thecurve for the 30.0 N constant tension case is also shown in the graphsfor comparison. The curve on the circumferential strain plot, whichcorresponds to the 30.0 to 20.0 N tension case, helps to illustrate theeffectiveness of the variable tension combined with a compliant layer.The strain in this case is relatively flat, which would correspond to amore uniform distribution of EFL along the length of a buffer tube.Refinements could be made to the tension curve, including non-linearvariations, to produce the optimum distribution of strain.

Additional cases were run using the same form of decaying tension, anddifferent values for the modulus of the compliant layer. A linearlydecaying tension from 30.0 to 20.0 N was chosen, and the modulus of thecompliant layer was varied from 0.08 to 0.13 GPa, in increments of 0.01GPa. The modulus of the regular material was 1.637 GPa, and Poisson'sratio for both materials was 0.1. Plots of the radial andcircumferential stress and strain are shown in FIG. 26B. Thecircumferential strain plot shows that the modulus of the compliantlayer can be tuned to achieve a strain distribution with littlevariation.

The effect of the shape of the decaying tension curve on the straindistribution was investigated using a series of curves. The tensioncurves were generated by the following equation: $\begin{matrix}{T = {T_{i} - {\left( {T_{f} - T_{i}} \right)\left\lbrack \frac{r - r_{o}}{R - r_{o}} \right\rbrack}^{\alpha}}} & (5.2)\end{matrix}$

where T_(i) is the starting tension, T_(f) is the final tension, r_(o)is the inner radius of the layers, R is the outer radius, and α is acoefficient influencing the shape of the curve. The starting tension wastaken to be 28.0 N, and the final tension was 20.0 N. Values for thecoefficient α were taken to be 0.4, 0.6, 1.0, 1.2, 1.6, and 2.0. Thetension curves for these values are shown in FIG. 27A.

The model with a compliant layer was used to run the different cases ofdecaying tension. The modulus of the compliant layer was 0.12 GPa, andthe modulus of the regular material was 1.637 GPa. The Poisson ratio forboth materials was 0.1. Plots of the radial and circumferential stressand strain are shown in FIG. 27B. The tension curve for the α=1.2 case,which is a slight deviation from linear, produces a circumferentialstrain with very little variation around a constant nominal value.

The results obtained using the concentric layer finite element modelwere compared to those obtained using the analytical model discussedpreviously. The material had a modulus of 1.637 GPa, and a Poisson'sratio of 0.1. A constant tension of 30.0 N was used for this case, andthere was no compliant layer on the reel surface. The analytical modeluses a parameter β to characterize the interaction between the layers ofmaterial and the reel. A few values of β were chosen to correspond tothe case of a rigid reel that was considered in the finite elementmodel. Plots of the radial and circumferential stress obtained from FEAand the analytical model are shown in FIG. 28. The stresses are in verygood agreement when the appropriate choice of β is made.

Comparisons were made between the computed EFL obtained using theconcentric layer finite element model and experimentally measuredvalues. The experimental data was obtained from a series of bufferingtrials of 2.5 mm diameter tubes. In each trial, a 12000.0 km length wastaken up on a reel that had an outer diameter of 401.7 mm, and a widthof 376.0 mm. The number of layers for this length of tube was determinedto be fifty-five from the following formula: $\begin{matrix}{L = {\sum\limits_{n = 1}^{N}{\frac{W}{d}\left\lbrack {d^{2} + {\pi^{2}\left( {D + d + {\left( {n - 1} \right)d\sqrt{3}}} \right)}^{2}} \right\rbrack}^{\frac{1}{2}}}} & (5.3)\end{matrix}$

where L is the length, W is the reel width, D is the reel diameter, d isthe buffer tube diameter, n is the layer number, and N is the totalnumber of layers. The equation for length was determined by assumingthat each buffer tube layer is wrapped around the reel in the path of ahelix, with a pitch equal to the buffer tube diameter. Also, perfectpacking of the tubes is assumed, as shown in FIG. 29.

A concentric layer model was created with an inner radius of 200.0 mm,and fifty-five layers of 2.5 mm thickness each. Since each solid layerof material represents a hollow buffer tube filled with gel and fibers,assigning the isotropic material properties for solid polypropylene toeach layer is not sufficient. An orthotropic material description wasused to allow for a softer modulus in the transverse direction of thetube. A local coordinate system was used to define the materialconstants for each layer. The local 1 direction was defined through thethickness of the layers, the local 2 direction was defined along thelength, and the local 3 direction was defined out of the plane. Thematerial properties were defined as follows:

E₁₁ = 0.24 GPa ν₁₂ = 0.09 G₁₂ = 0.60 GPa E₂₂ = 1.20 GPa ν₂₃ = 0.09 G₂₃ =0.60 GPa E₃₃ = 0.24 GPa ν₃₁ = 0.09 G₃₁ = 0.60 GPa

A simulation was performed for the case of 30.0 N constant tension, andthe EFL was computed. The EFL was computed from the circumferentialstrain according to Equation 5.1, with an initial EFL value of 1.0%. Theinitial EFL for the buffering trials was not known so the value wasapproximated to achieve a reasonable agreement with the experiment. Thecomputed and the experimentally measured EFL distribution are shown inFIG. 30. The length scale in the plot has been normalized to one. Theresults show good agreement between model and experiment except for aslight deviation on the right portion of the curves. The EFL valuesgenerated by the model are computed from a single strain point in eachlayer, so the values are only a representation of what the distributionmay look like along the length of a buffer tube. Also, the experimentalmeasurement technique can have some inaccuracy due to the way the EFL issampled and the individual measurements are made. The technique consistsof cutting the tube into short sections at discrete locations, andmeasuring the length of the fiber in each section. The shape of the EFLcurve depends on the frequency of the sampling, and the accuracy of thehandling and measurement of each section.

The numerical simulations did not take any thermal effects intoconsideration for the computation of strain. Thermal and materialeffects, such as expansion and contraction, and material crystallinityand shrinkage, influence the distribution of EFL within the tube. Also,relaxation of the material while it is on the reel has an effect on theEFL. In addition, the EFL curve predicted by the model represents theEFL while the buffer tube is still on the reel in a strained state. Themeasurements are taken when the buffer tube has been removed from thereel, and cut into pieces. There is the possibility that the EFL couldchange in this situation, and the values could not be directly comparedto those predicted by the model. It is assumed that the EFL is locked inwhile the buffer tube is on the reel, and it does not change much whenit is taken off, the model provides a reasonable approximation. Variousthermal and material effects can be taken into consideration in themodel, assuming the appropriate material data can be provided.

Another simulation was performed to compare with an experimentalbuffering trial conducted with a constant tension of 10.0 N. In thiscase the shape of the EFL curve predicted by the model is not the sameas the experiment, so the initial EFL value can not be determined bymatching the curves. FIG. 31 shows the EFL curve from the experiment andfrom the model, calculated with an initial EFL of 0.52%. The model showsthat the entire EFL curve shifts up due to less tension on the tube.This is reasonable since less strain on the tube would allow more EFL toaccumulate, and would not greatly reduce the EFL already present. Theexperimentally measured EFL shows a very low EFL in the tube near thesurface of the reel. This level of EFL seems to indicate that a tensionhigher than 10.0 N was on the tube at the beginning of the trial. Theprocess may be slightly unstable at the beginning when the first fewlayers are going onto the reel, and transients are still present. Also,it is possible that for very low tensions the thermal effects on thematerial become more significant.

FIG. 32A shows a comparison of the EFL for the 10.0 N and 30.0 N tensioncases, for both the model and experiment. The comparison between FEA andexperiment shows the same trend in EFL except for the beginning portionof the experimental curve for the 10.0 N case.

Another case that was compared included a thick compliant layer on thereel surface, and a linearly decaying tension. The material propertiesfor the regular material were taken to be the same as in the simulationsdiscussed previously, and the compliant layer had a modulus of 12.0 MPa,and a Poisson's ratio of 0.09. The first two layers of elements weretaken to be the compliant material. The tension was decayed linearlyfrom a starting value of 28.0 N to 11.0 N. FIG. 32B shows the EFLobtained from the model, and the experimentally measured distribution.An initial EFL of 0.65% was chosen for the model calculations. The trendof EFL is predicted quite well in this case, although there is a slightdeviation from the measured values. There are several uncertainties inthis case that could result in some differences in the measured andpredicted EFL values. The properties of the actual compliant layer werenot known, and the assumption of it being a linear elastic material maynot be sufficient. Also, the tension in the experiment was adjustedmanually in a step-wise manner, which may have introduced transientsinto the system and could have caused a deviation from the linear decaycurve assumed.

The comparisons made between the model and the experiments showreasonably good agreement in the trend of the EFL distribution. Thisallows the model to be used in a predictive capacity to help determinethe most favorable conditions for obtaining a uniform EFL distribution.Although there may be some uncertainties in material properties or otherprocess parameters, the model can be used to help bracket a solution.The numerical simulations could be supplemented with carefullycontrolled experiments to help tune the solution further.

To validate and confirm the analysis performed, and presented above,various experiments were performed. The experiments were directed todetermining the influence of a number of factors on EFL, including theinfluence of time on the reel, use of foam pads, constant and variabletake-up tension, and variation in the line speed or angular velocity ofthe spool on the distribution of EFL in buffer tubes.

One of the first experiments conducted was the influence of the time thebuffer tube stands on the reel after manufacture. Analysis of the EFLdistribution in three 3-km long buffer tubes was performed. Results aresummarized in FIG. 33 and Table I (shown below). The end points of lines1*, 2*, and 3* were obtained the same day that the tubes werefabricated. Curves 1*-3* show the EFL distribution that was obtainedwhen the tubes were cut and measured ten days later.

The curve 1* in FIG. 33 was obtained from a buffer tube that was notreeled on the spool. Instead, the buffer tube was placed in a box whereit cooled to the room temperature. Variation of EFL in this case is from0.60% to 0.69% without a clear “parabolic” distribution typical forreeled buffer tube.

The curve 2* corresponds to the case when the buffer tube was placed inthe box for 7 days, then reeled under 1.5 kg tension on the reel, kepton the reel for 3 days, then unwrapped for an EFL measurement. Thiscurve has features of a parabolic shape with 0.38% EFL at the beginningof the buffer tube, up to 0.57% in the middle, and 0.34% on the end ofthe buffer tube. Curve 2* is located below Curve 1*. The reduction inthe levels of EFL is apparently due to the action of the circumferentialforces of tension in the reeled buffer tube causing 3-day elongation(creep) of the polymeric material with respect to the fibers.

The parabolic-type curve 3* is located below curves 1* and 2*,apparently due to increased time (10 days) of stretching of the rolledbuffer tube by circumferential stresses. The EFL values are ranging from−0.02% EFL at the beginning of the buffer tube, to 0.24% in the middle,and 0.10% on the end of the buffer tube.

Comparison of the curves obtained the same day and 10 days afterfabrication of buffer tubes suggested the following. In the reeled tubesEFL reduced while in unreeled tubes EFL increased in time. This can berelated to the thermal cooling and shrinkage of thermoplastic materials;in the reeled tubes the shrinkage is restricted by existingcircumferential stresses. When the contribution from the stresses ishigher than that of residual shrinkage, relative elongation ofthermoplastic materials is higher than shrinkage. Consequently, theresulting elongation would result in a reduction of EFL. In contrast, inunreeled buffer tubes, the residual shrinkage is not restrained andfinal values of EFL increase.

TABLE I Measured Values of EFL in Three Buffer Tubes. Average AverageAverage Average Measure- Sample EFL in EFL in EFL in EFL in AverageAverage Average Average ment time Location Tube 0 Tube 1 Tube 2 Tube 3EFL in EFL in EFL in EFL in Sample (days) (m) (mm) (mm) (mm) (mm) Tube 0Tube 1 Tube 2 Tube 3 OSE 0 3000 18.6 16.0 10.4 0.61% 0.52% 0.34% ISE-1 00 4.1 14.5 14.7 0.13% 0.47% 0.48% ISE-2 7 0 5.9 0.19%  1 10 0 19.8 11.5−0.6 0.65% 0.38% −0.02%  2 10 300 20.5 13.2 2.3 0.67% 0.43% 0.08%  3 10600 20.5 14.0 5.2 0.67% 0.46% 0.17%  4 10 900 18.3 13.9 3.3 0.60% 0.46%0.11%  5 10 1200 19.8 16.0 6.5 0.65% 0.53% 0.21%  6 10 1500 19.5 16.47.3 0.64% 0.54% 0.24%  7 10 1800 20.4 17.4 7.3 0.67% 0.57% 0.24%  8 102100 18.4 16.6 7.1 0.60% 0.54% 0.23%  9 10 2400 20.1 13.4 6.2 0.66%0.44% 0.20% 10 10 2700 19.9 11.8 7.8 0.65% 0.39% 0.26% 11 10 3000 20.910.5 3.1 0.69% 0.34% 0.10%

Monotonic reduction in take-up tension should result in a flatterdistribution of circumferential stresses in the roll. Consequently,changes in EFL are expected to be within a narrower range as compared tothe case of constant take-up tension. In addition, finite elementmodeling showed that adding a soft foam pad on the core or periodicallyinserting soft pads into the roll should increase the range of variationof EFL.

Experiments were performed on the 12-km buffer tubes. The first buffertube was wrapped around a spool at a constant take-up tension. Thecorresponding EFL curve is shown in FIG. 34. The second buffer tube wasreeled on the same spool but with a double-layer of thick foam on thecore. Also, in the case of the second buffer tube, the take-up tensionwas monotonically reduced from 25N to approximately 9N. Thecorresponding EFL curve is shown in FIG. 34. As can be seen from FIG.34, the parabolic curve typical for reeled buffer tubes on a bare reelis actually shallower than the curve obtained with a pad and variabletake-up tension. Based on these results, it was suggested to furtherstudy the possible nonlinear effect of a thinner soft pad on initialvalues of EFL toward the goal of obtaining a flatter curve.

FIG. 35 illustrates an approach using a thin foam layer on the“regular-rigidity” core and decaying take-up tension to minimize thevariation in the EFL values.

Typically, reeling is performed at a constant line speed, i.e. constantangular velocity of the rotating spool. As described previously, avariation in the line speed and the corresponding variation in angularvelocity of the reel produces variations along the radius of the buffertube roll in temperature and possibly tensile load. This resulted in theconcept of using a monotonically variable angular velocity of the spoolto control the stress state in the buffer tubes and subsequently theEFL.

Experiments were focused on a three-step angular velocity process. Thefirst step is an initial stage of the reeling process when angularvelocity is increased from zero to a prescribed value. The second stepis ramping or transition in angular velocity from 100 m/min to 400m/min. The third step is a non-ramping scenario to the end of thereeling process, when the angular velocity is kept constant. Thesestages of the reeling process are depicted in FIG. 36, where the angularvelocity, ω, is related to the linear velocity, v, of the buffer tubeand current radius, r, as shown below:

Results of several experiments (Exp. 14, Exp. 15 and Exp. 16) aresummarized in FIG. 37, which shows EFL distribution as a function oflength of the buffer tube. In all three cases shown, a thin soft pad onthe reel core and decaying take-up tension were used. These resultssuggested the influence of the variable angular velocity on the EFLcurves. Curve 14 is obtained at a relatively high constant linear speedof 400 m/min. As previously discussed, a high level of line speedreduces cooling time for the buffer tubes and reduces thetime-to-stretch (creep, reduction in the Young's modulus), andconsequently, produces relatively high levels of EFL.

Also, a transition from lower to higher line speed increases the coolingtime for the initial part of buffer tubes, increases the time-to-stretch(creep, reduction in the Young's modulus), and consequently, reduceslevels of EFL at the beginning of the buffer tube. Further, a dynamictransition from lower to higher speed adds inertia forces of tension andthus increases stretching of thermoplastic material and reduces EFL inthe initial part of buffer tube length. Curve 15 is obtained via rampingwhen the linear velocity was monotonically increased (as a linearfunction) from an initial value of 100 m/min to 400 m/min, and achievedits maximum of 400 m/min when the tube length was about 1.5 km (dashedline in FIG. 37).

Constant lower line speed uniformly increases the cooling time,increases the time-to-stretch, and consequently, uniformly reduceslevels of EFL. Curve 16 is obtained at a constant line speed of 100m/min.

To further analyze one embodiment of the present invention, furtheranalysis using thin foam pads on reels and monotonically decaying thetake-up tension was investigated. For this purpose, a system based on abucket of water and a valve was built and successfully used. It wasfound that this system provided results with good repeatability. Thevalve was used to accelerate water release to provide a parabolic decayin the tension. Friction of the bucket against a pole additionallyprovided a favorable reduction in the vibration of the load andpresumably, smoother EFL curves.

FIG. 38 shows a modified line according to the present invention thatemployed a bucket of water to control take-up tension in the form of aparabolically decaying function. FIG. 39 depicts the EFL curves for twoexperiments using the tension control. The case corresponding to curveExp. 27 was performed with the following sequence of take-up load: startwith 30N, after the first 4000 m the load is 27N, after 8500 m the loadis 20N, then it decreases monotonically down to 12N. One thin layer offoam was wrapped around a regular steel core, and the line speed waskept constant at 400 m/min. Curve Exp. 31 represents another scenario:one thin layer of foam wrapped around a regular steel core, with aninitial line speed of 350 m/min, and a take-up tension of 30 N. After 9km of buffer tube was made, the flow rate of water was increased.

The main result of the experiments involving a bucket of water with avalve is controllability of the EFL using soft foam pads and decayingtake-up tension. Based on the experiments, it is recommended that apneumatic-controlled system for more accurate computer-controlledtake-up tension is to be used to obtain a constant-value EFL, althoughany system capable of provide the control needed can be used.

In reviewing the above analysis and experimentation a number ofembodiments of the present invention are contemplated, where variousaspects of the buffer tube manufacturing process are used individually,or in combination, to achieve a buffer tube or cable having asubstantially even EFL distribution along its length.

In a first embodiment of the present invention, the take-up tension ofthe buffer tube is monotonically decayed as the tube is wound on thereel. The exact function used to decay the tension would be governed bythe individual characteristics of the manufacturing system to be used,but is to be optimized by taking into account all of the factorspreviously discussed, including, line speed, reel core diameter,material properties, etc. Although it is preferred that a monotonicallydecaying function be used, it is contemplated that other functions mayalso be used to decay the tensile load on the tube during manufacture,without expanding the scope or spirit of the present invention. Further,although one of the main purposes of the present invention is to createuniform EFL distribution throughout the length of a buffer tube, it iscontemplated that the present invention can be used to create acontrolled non-uniform EFL distribution throughout the length of thecable, where such a non-uniform distribution is desired.

In the preferred embodiment, the decaying tension is to be supplied by apneumatic-controlled system for more accurate computer-controlledtake-up tension and to obtain a constant-value EFL. However, any knownor commonly used system that can provide the same control can be used,such as mechanical or hydraulic, which is capable or providing afunctionally changing tensile force to the buffer tubes, as they arebeing wound.

Further, in this embodiment it is preferred that the reel core becovered with a stiffness-compliant pad or sleeve, such as a thinpackaging foam pad, as previously discussed, to provide stress relief inthe initial layers of the buffer tubes located near the reel coresurface (as the windings of the tube closer to the reel core surfacehave a high increase in EFL). In the preferred embodiment, thethickness, porosity and Young's modulus of the pad is selected toprovide the desired stress absorption in the inner layers and provide asubstantially uniform EFL distribution in the tube. As with the decayfunction used, the pad used will be governed by the particularcharacteristics and design parameters of the manufacturing process.

In a second embodiment of the present invention, the angular velocity ofthe rotating take-up reel is varied in accordance with a monotonicalfunction (similar to that in the first embodiment with regard to thedraw tension) to provide a substantially uniform EFL distribution alongthe length of the tube. As with the first embodiment, the exact functionused to decay the angular velocity would be governed by the individualcharacteristics of the manufacturing system to be used, but is to beoptimized by taking into account all of the factors previouslydiscussed, including, line tension, reel core diameter, materialproperties, etc. Although it is preferred that a monotonical function beused, it is contemplated that other functions may also be used to varythe reel angular velocity during manufacture, without expanding thescope or spirit of the present invention. Further, although one of themain purposes of the present invention is to create uniform EFLdistribution throughout the length of a buffer tube, it is contemplatedthat the present invention can be used to create a controllednon-uniform EFL distribution throughout the length of the cable, wheresuch a non-uniform distribution is desired.

It should be noted that unlike in the first embodiment, to achieve asubstantially uniform EFL using the second embodiment of the presentinvention the function used should increase the angular speed over timethrough ramping (unlike the first embodiment which decreased the tensionover time).

In the preferred embodiment, the angular speed variation can be providedby any known or commonly used system that can provide adequate controlof the required speed variations to ensure the function chosen tocontrol the speed is followed as accurately as possible. Existing cablemanufacturing devices can be modified to change the speed of the buffertube and reeling system to adjust the angular velocity of the take-upreel.

Further, in this embodiment it is preferred that the reel core becovered with a stiffness-compliant pad or sleeve, such as a thinpackaging foam pad, as previously discussed, to provide stress relief inthe initial layers of the buffer tubes located near the reel coresurface (although as with the first embodiment the use of the pad is notnecessary). As with the function used, the pad used will be governed bythe particular characteristics and design parameters of themanufacturing process.

It should be noted that it is contemplated that a combination of theabove two embodiments (use of monotonically decreased tension withvariations of the reel angular speed) can be used to achieve asubstantially uniform EFL distribution.

In a third embodiment of the present invention, soft-cushion pads areinserted periodically with the tubes during the reeling process. This isdepicted in FIG. 40 where a pad 41 is periodically inserted in thewinding of the tube 42. It is preferred that the pad 41 be the width ofthe spool 44 and a length equal to the spool circumference at the pointof insertion. It is also preferred that the pad be of a material, suchas foam sheets, which would accommodate shrinkage in the tube as itcools on the reel 44. It should be noted that as an alternative to usingperiodically inserted pads 41, it is contemplated that a continuous padbe fed with the buffer tube 42 onto the reel 44 so that a cushioninglayer is provided continuously throughout the reeled buffer tube 42. Itis further contemplated that a core pad 45, as previously describedabove, can also be used in this embodiment.

The use of the periodic insertion of pads 41 according to the presentembodiment provides space to accommodate shrinkage in the layers of tube42, so the wraps of the tube 42 are allowed to move in the radialdirection and slide toward the center of the spool under the residualloads. The pads 42 act as energy absorbing elements and deform duringthe cooling of the tube 42 when additional stresses from the tubecontraction (during cooling) are experienced. Further, the pads 42 actas spacers to reduce cumulative changes in stresses in very long buffertube manufacturing. Instead of a long single tube (which was shownpreviously as having adverse effects on EFL distributions) the padseffectively create a set of smaller reeled windings separated by thecushions or pads 41.

The pads 41 to be used are to have a thickness, porosity and Young'smodulus optimized for the particular manufacturing system andspecifications and should be optimized to produce a substantiallyuniform EFL distribution throughout the length of the tube 42. It shouldbe noted that although it is preferred that the pads 41 be of a softermaterial (having a Young's modulus less than that the of the tube 42) toallow distortion under the loads on the reel, it is also contemplatedthat a series of stiff panels or planks 43 can be periodically insertedinto the tube 42 winding. The panels inserted into the windings are aswide as the spool or reel such that the panels 43 rest on shelves (notshown) on the inner surfaces of the edges of the spools 44. It ispreferred that, these layers or planks 43 are pre-curved to avoid pointcontacts with the tubes and should be made of a material with a Young'smodulus higher than that of the buffer tube 42 (unlike the pads 41). Thepanels 43 act as a shelf separating various groupings of windings of thetube 42 thus avoiding the effects of the compounding stresses of asingle wound tube (previously shown and discussed).

In this embodiment, it is also preferred to re-reel the tube 42 onto adifferent reel after the tube 42 manufacture is complete, and it isallowed to cool to room temperature. -It is preferred that the pads 41be removed during this step to achieve optimum winding of the tubes 42as they are re-wound on a second reel (not shown).

It is noted that it is contemplated that the above embodiment may becombined with either of the previously discussed embodiments to providea substantially uniform EFL distribution, or an EFL distributionaccording to desired specifications. Further, it is contemplated thatany of the aspects of the above embodiments may be combined in part ortotally to achieve the desired EFL distributions.

Finally, it is noted that to obtain a substantially even EFLdistribution other parameters can be optimized, such as increasing thediameter of the reel core to a relatively large starting size reducingthe overall length of the manufactured cable to reduce the number ofwraps which can create large combined residual loads and more intensivecooling of the tube prior to the spooling of the tube. It is importantto note however, that the same dimensions may not be applicable in allcases, and the parameters may need to be adjusted and optimizeddepending on the materials and tube sizes used.

FIG. 41 shows a schematic diagram of an apparatus or system that can beused to perform the method of the present invention. In the systemshown, the optical fibers 49 are taken off of their respective payoffreels 48 and drawn through an extruder 50 which places the fibers 49 ina buffer tube 42 which is wound. It is noted that the fibers 49 showncan also be fiber optic ribbons or any other material or grouping to bewound. Further, it is noted that any commonly known devices andcomponents can be used to perform these functions. After the buffertubes 42 are extruded they are passed through an optional cooling deviceor system 51, which can be any commonly known or used cooling device.After the tube 42 is cooled the EFL of the tube 42 is measured by an EFLmeasuring device 52 which can be any commercially available EFLmeasurement device. After the EFL measurement, the tube passes through abuffer tube tensioner 53. The buffer tube tensioner applies a variable,i.e. non-constant tensile load onto the buffer tube according a desiredfunction, such as monotonically decaying or parabolic function. In thepreferred embodiment the tensioner is pneumatic and computer controlledso as to accurately control the tensile load on the buffer tube 42.However, the tensioner 53 can also be hydraulic or mechanical, as longas it is capable of functionally changing the tensile load on the buffertube 42 in accordance with a desired function. After the tube 42 passesthrough the tensioner 53 it is wound on a spool 44, having a stiffnesscompliant pad 45 in accordance with the present invention. The use ofthe pad 45 is not necessary but is preferred. Further, in a preferredembodiment, a pad inserter 55 places either an additional pad 41 orplanks 43 onto the spool 44 with the buffer tube 42 to aid in reducingthe EFL. The inserter 55 can be any commonly known or used payout deviceused to place a material onto a wound spool and can be positioned at anyreasonable position regarding the location of the spool 44. The angularvelocity of the spool 44 is controlled by an angular velocity controller54. The controller 54 is capable of controlling the angular velocity ofthe spool 44 in accordance with a desired or programmed function tooptimize the operation of the system. Finally, in a preferred embodimentof the present invention, a second spool 44′ is positioned near theprimary spool 44 to allow the buffer tube 42 to be re-wound onto thesecond spool 44′. The second spool 44′ can be either controlled by thesame angular velocity control 54 (as shown in FIG. 41) or can becontrolled by its own independent control (not shown). Further, the pad41 or planks 43 can either be removed between the first spool 44 andsecond spool 44′ or can be re-wound with the buffer tube 42 depending onthe manufacture requirements or specifications of the system. It shouldalso be noted that the second spool 44′ can have a buffer pad (notshown) on its core to aid in controlling the EFL of the buffer tube 42.

Further, it should be noted that experiments have shown that heatedthermoplastic tubes can be easily and permanently stretched after thetubes are heated. This can be done by using any existing heatingtechniques including heat radiation from a tubular heater 56 placedbetween the first 44 and second 44′ spools. This procedure would be usedin a situation where after the initial spooling of the buffer tubes iscompleted, and an EFL measurement is made on the spooled tubes, it isfound that the EFL of the tube is still not at an acceptably high levelor having large EFL variation along the tube length. Instead ofscrapping the spooled buffer tubes they can be “re-spooled” onto asecond spool 44′ while a heater 56 (placed between the spools) heats thebuffer tube 42 to allow it to stretch to correct the EFL error thatexisted in the tube after the first spooling. During the re-spooling andheating of the buffer tubes, the tubes can have an tensile load appliedto them in accordance with a method described previously, where forexample a second monotonically decaying function is used and applied asthe buffer tube is being re-spooled through a second buffer tubetensioner 53′. The function used may be the same or different than thatused for the first spooling of the buffer tube and the same or differentmethodology may be applied in the re-spooling process. For example, thetension on the tube may be monotonically decayed during the initialspooling of the buffer tube 42, and during the re-spooling of the buffertube (if necessary) the angular velocity of the second spool may befunctionally changed (instead of the tension applied) to correct anyexisting EFL error depending on the amount and type of correctionneeded. It should be noted that the present invention is not limited tothe above example, and any combination of the previously discussedmethods may be used to correct the tube EFL during the second spoolingstage.

These additional features of the present invention are optional and maynot be needed in all circumstances, depending on the manufacturing andproduction needs and characteristics. The heater 56 and secondarytensioner 53′ are optional and are not required. Additionally, a singleangular velocity controller 54 may be used to control both spools 44,44′, or individual controls may be used if such a configuration is morefeasible. Additionally, it may be beneficial to use a stiffnesscompliant pad 45 on both the first 44 and secondary 44′ spool dependingon the methodology used to correct any error in buffer tube EFL.

Further, it is also known from experiments, that multiple re-spoolingeven under constant tension and ambient temperature often improveoptical performance of the cables and buffered fibers presumably due toreduction in the micro-stress and micro-adjustment of the fiberposition. Therefore, the present invention is not limited to only afirst re-spooling of the tubes, but may be used with a multiplere-spooling where some of the disclosed methods of correcting EFL(discussed previously) may or may not be used.

It is of course understood that departures can be made from thepreferred embodiments of the invention by those of ordinary skill in theart without departing from the spirit and scope of the invention that islimited only by the following claims.

It is further understood that the present invention is not limited tothe manufacture of fiber optic buffer tubes, but can be applied to anyother industry or application where long lengths of material is wound onspools or reels and it is desired to control, or reduce the adverseeffects of, residual stresses and strains in the rolled or woundmaterials.

I claim:
 1. An apparatus for winding a material; comprising: a spoolhaving a core and a buffer pad on an outer surface of said core; apayout device to payout said material to be wound; and a tensionerapplying a draw tension to said material upstream of said spool, whereinsaid tensioner functionally changes said draw tension of said materialas said material is being wound onto said spool.
 2. The apparatusaccording to claim 1, wherein said tensioner monotonically changes saiddraw tension during winding of said material.
 3. The apparatus accordingto claim 1, wherein said buffer pad has a Young's modulus lower thanthat of said material.
 4. The apparatus according to claim 1, furthercomprising a driver connected to said spool and said driver varies anangular speed of said spool while said material is winding onto saidspool.
 5. The apparatus according to claim 4, wherein said angular speedis varied by said driver according to a monotonical function.
 6. Theapparatus according to claim 1, further comprising an inserter insertinga second pad between windings of said material as said material is beingwound.
 7. The apparatus according to claim 6, wherein said second padhas a Young's modulus lower than that of said material.
 8. The apparatusaccording to claim 1, further comprising a second spool onto which saidmaterial is wound after said winding onto said spool.
 9. The apparatusaccording to claim 8, wherein said second spool has a second core ontowhich a second buffer pad is positioned.
 10. An apparatus for winding afiber optic buffer tube; comprising: a spool having a core and a bufferpad on an outer surface of said core; a payout device to payout saidbuffer tube to be wound; and a tensioner applying a draw tension to saidbuffer tube upstream of said spool, wherein said tensioner functionallychanges said draw tension of said buffer tube as said buffer tube isbeing wound onto said spool.
 11. The apparatus according to claim 10,wherein said tensioner monotonically changes said draw tension duringwinding of said buffer tube.
 12. The apparatus according to claim 10,wherein said buffer pad has a Young's modulus lower than that of saidbuffer tube.
 13. The apparatus according to claim 10, further comprisinga driver connected to said spool and said driver varies an angular speedof said spool while said buffer tube is winding onto said spool.
 14. Theapparatus according to claim 13, wherein said angular speed is varied bysaid driver according to a monotonical function.
 15. The apparatusaccording to claim 10, further comprising an inserter inserting a secondpad between windings of said buffer tube as said buffer tube is beingwound.
 16. The apparatus according to claim 15, wherein said second padhas a Young's modulus lower than that of said buffer tube.
 17. Theapparatus according to claim 10, further comprising a second spool ontowhich said buffer tube is wound after said winding onto said spool. 18.The apparatus according to claim 17, wherein said second spool has asecond core onto which a second buffer pad is positioned.
 19. Theapparatus of claim 17, further comprising a buffer tube heater forheating said buffer tube while said buffer tube is being wound onto saidsecond spool.
 20. The apparatus of claim 17, further comprising a secondtensioner applying a second draw tension to said buffer tube while saidbuffer tube is winding onto said second spool.
 21. The apparatus ofclaim 17, further comprising an inserter inserting a second pad betweenwindings of said buffer tube as said buffer tube is being wound ontosaid second spool.
 22. The apparatus of claim 17, further comprising adriver connected to said second spool and said driver varies an angularspeed of said second spool while said buffer tube is winding onto saidsecond spool.
 23. The apparatus according to claim 22, wherein saidangular speed is varied by said driver according to a monotonicalfunction.